{"id":8178,"date":"2024-04-27T06:00:32","date_gmt":"2024-04-27T04:00:32","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=8178"},"modified":"2024-04-26T16:53:05","modified_gmt":"2024-04-26T14:53:05","slug":"27-abr-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/27-abr-24\/","title":{"rendered":"La semana en Calculemus (27 de abril de 2024)"},"content":{"rendered":"\n<p>Esta semana he publicado en <a href=\"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/\">Calculemus<\/a> las demostraciones con Lean4 e Isabelle\/HOL de las siguientes propiedades:<\/p>\n<ul>\n<li><a href=\"#ej1\">1. Uni\u00f3n con la imagen<\/a><\/li>\n<li><a href=\"#ej2\">2. Intersecci\u00f3n con la imagen<\/a><\/li>\n<li><a href=\"#ej3\">3. Uni\u00f3n con la imagen inversa<\/a><\/li>\n<li><a href=\"#ej4\">4. Imagen de la uni\u00f3n general<\/a><\/li>\n<li><a href=\"#ej5\">5. Imagen de la intersecci\u00f3n general<\/a><\/li>\n<\/ul>\n<p>A continuaci\u00f3n se muestran las soluciones.<br \/>\n<!--more--><br \/>\n<a name=\"ej1\"><\/a><\/p>\n<h3>1. Uni\u00f3n con la imagen<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ f[s \u222a f\u207b\u00b9[v]] \u2286 f[s] \u222a v &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable (\u03b1 \u03b2 : Type _)\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s : Set \u03b1)\nvariable (v : Set \u03b2)\n\nexample : f '' (s \u222a f \u207b\u00b9' v) \u2286 f '' s \u222a v :=\nby sorry\n<\/pre>\n<h4>1.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Sea &#92;(y \u2208 f[s \u222a f\u207b\u00b9[v]]&#92;). Entonces, existe un x tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s \u222a f\u207b\u00b9[v] &#92;tag{1} &#92;&#92;<br \/>\n   &amp;f(x) = y       &#92;tag{2}<br \/>\n&#92;end{align}<br \/>\nDe (1), se tiene que &#92;(x \u2208 s&#92;) \u00f3 &#92;(x \u2208 f\u207b\u00b9[v]&#92;). Vamos a demostrar en ambos casos que<br \/>\n&#92;[ y \u2208 f[s] \u222a v &#92;]<\/p>\n<p><strong>Caso 1<\/strong>: Supongamos que &#92;(x \u2208 s&#92;). Entonces,<br \/>\n&#92;[ f(x) \u2208 f[s] &#92;]<br \/>\ny, por (2), se tiene que<br \/>\n&#92;[ y \u2208 f[s] &#92;]<br \/>\nPor tanto,<br \/>\n&#92;[ y \u2208 f[s] \u222a v &#92;]<\/p>\n<p><strong>Caso 2<\/strong>: Supongamos que &#92;(x \u2208 f\u207b\u00b9[v]&#92;). Entonces,<br \/>\n&#92;[ f(x) \u2208 v &#92;]<br \/>\ny, por (2), se tiene que<br \/>\n&#92;[ y \u2208 v &#92;]<br \/>\nPor tanto,<br \/>\n&#92;[ y \u2208 f[s] \u222a v &#92;]<\/p>\n<h4>1.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable (\u03b1 \u03b2 : Type _)\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s : Set \u03b1)\nvariable (v : Set \u03b2)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (s \u222a f \u207b\u00b9' v) \u2286 f '' s \u222a v :=\nby\n  intros y hy\n  obtain \u27e8x : \u03b1, hx : x \u2208 s \u222a f \u207b\u00b9' v \u2227 f x = y\u27e9 := hy\n  obtain \u27e8hx1 : x \u2208 s \u222a f \u207b\u00b9' v, fxy : f x = y\u27e9 := hx\n  cases' hx1 with xs xv\n  . -- xs : x \u2208 s\n    have h1 : f x \u2208 f '' s := mem_image_of_mem f xs\n    have h2 : y \u2208 f '' s := by rwa [fxy] at h1\n    show y \u2208 f '' s \u222a v\n    exact mem_union_left v h2\n  . -- xv : x \u2208 f \u207b\u00b9' v\n    have h3 : f x \u2208 v := mem_preimage.mp xv\n    have h4 : y \u2208 v := by rwa [fxy] at h3\n    show y \u2208 f '' s \u222a v\n    exact mem_union_right (f '' s) h4\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (s \u222a f \u207b\u00b9' v) \u2286 f '' s \u222a v :=\nby\n  intros y hy\n  obtain \u27e8x : \u03b1, hx : x \u2208 s \u222a f \u207b\u00b9' v \u2227 f x = y\u27e9 := hy\n  obtain \u27e8hx1 : x \u2208 s \u222a f \u207b\u00b9' v, fxy : f x = y\u27e9 := hx\n  cases' hx1 with xs xv\n  . -- xs : x \u2208 s\n    left\n    -- \u22a2 y \u2208 f '' s\n    use x\n    -- \u22a2 x \u2208 s \u2227 f x = y\n    constructor\n    . -- \u22a2 x \u2208 s\n      exact xs\n    . -- \u22a2 f x = y\n      exact fxy\n  . -- \u22a2 y \u2208 f '' s \u222a v\n    right\n    -- \u22a2 y \u2208 v\n    rw [\u2190fxy]\n    -- \u22a2 f x \u2208 v\n    exact xv\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (s \u222a f \u207b\u00b9' v) \u2286 f '' s \u222a v :=\nby\n  rintro y \u27e8x, xs | xv, fxy\u27e9\n  -- y : \u03b2\n  -- x : \u03b1\n  . -- xs : x \u2208 s\n    -- \u22a2 y \u2208 f '' s \u222a v\n    left\n    -- \u22a2 y \u2208 f '' s\n    use x, xs\n    -- \u22a2 f x = y\n    exact fxy\n  . -- xv : x \u2208 f \u207b\u00b9' v\n    -- \u22a2 y \u2208 f '' s \u222a v\n    right\n    -- \u22a2 y \u2208 v\n    rw [\u2190fxy]\n    -- \u22a2 f x \u2208 v\n    exact xv\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (s \u222a f \u207b\u00b9' v) \u2286 f '' s \u222a v :=\nby\n  rintro y \u27e8x, xs | xv, fxy\u27e9 <;>\n  aesop\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (t : Set \u03b1)\n-- #check (mem_image_of_mem f : x \u2208 s \u2192 f x \u2208 f '' s)\n-- #check (mem_preimage : x \u2208 f \u207b\u00b9' v \u2194 f x \u2208 v)\n-- #check (mem_union_left t : x \u2208 s \u2192 x \u2208 s \u222a t)\n-- #check (mem_union_right s : x \u2208 t \u2192 x \u2208 s \u222a t)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Union_con_la_imagen.lean\">Lean 4 Web<\/a>.<\/p>\n<h4>1.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Union_con_la_imagen\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (s \u222a f -` v) \u2286 f ` s \u222a v\"\nproof (rule subsetI)\n  fix y\n  assume \"y \u2208 f ` (s \u222a f -` v)\"\n  then show \"y \u2208 f ` s \u222a v\"\n  proof (rule imageE)\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s \u222a f -` v\"\n    then show \"y \u2208 f ` s \u222a v\"\n    proof (rule UnE)\n      assume \"x \u2208 s\"\n      then have \"f x \u2208 f ` s\"\n        by (rule imageI)\n      with \u2039y = f x\u203a have \"y \u2208 f ` s\"\n        by (rule ssubst)\n      then show \"y \u2208 f ` s \u222a v\"\n        by (rule UnI1)\n    next\n      assume \"x \u2208 f -` v\"\n      then have \"f x \u2208 v\"\n        by (rule vimageD)\n      with \u2039y = f x\u203a have \"y \u2208 v\"\n        by (rule ssubst)\n      then show \"y \u2208 f ` s \u222a v\"\n        by (rule UnI2)\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (s \u222a f -` v) \u2286 f ` s \u222a v\"\nproof\n  fix y\n  assume \"y \u2208 f ` (s \u222a f -` v)\"\n  then show \"y \u2208 f ` s \u222a v\"\n  proof\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s \u222a f -` v\"\n    then show \"y \u2208 f ` s \u222a v\"\n    proof\n      assume \"x \u2208 s\"\n      then have \"f x \u2208 f ` s\" by simp\n      with \u2039y = f x\u203a have \"y \u2208 f ` s\" by simp\n      then show \"y \u2208 f ` s \u222a v\" by simp\n    next\n      assume \"x \u2208 f -` v\"\n      then have \"f x \u2208 v\" by simp\n      with \u2039y = f x\u203a have \"y \u2208 v\" by simp\n      then show \"y \u2208 f ` s \u222a v\" by simp\n    qed\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (s \u222a f -` v) \u2286 f ` s \u222a v\"\nproof\n  fix y\n  assume \"y \u2208 f ` (s \u222a f -` v)\"\n  then show \"y \u2208 f ` s \u222a v\"\n  proof\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s \u222a f -` v\"\n    then show \"y \u2208 f ` s \u222a v\"\n    proof\n      assume \"x \u2208 s\"\n      then show \"y \u2208 f ` s \u222a v\" by (simp add: \u2039y = f x\u203a)\n    next\n      assume \"x \u2208 f -` v\"\n      then show \"y \u2208 f ` s \u222a v\" by (simp add: \u2039y = f x\u203a)\n    qed\n  qed\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (s \u222a f -` v) \u2286 f ` s \u222a v\"\nproof\n  fix y\n  assume \"y \u2208 f ` (s \u222a f -` v)\"\n  then show \"y \u2208 f ` s \u222a v\"\n  proof\n    fix x\n    assume \"y = f x\"\n    assume \"x \u2208 s \u222a f -` v\"\n    then show \"y \u2208 f ` s \u222a v\" using \u2039y = f x\u203a by blast\n  qed\nqed\n\n(* 5\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (s \u222a f -` u) \u2286 f ` s \u222a u\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej2\"><\/a><\/p>\n<h3>2. Intersecci\u00f3n con la imagen<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ f[s] \u2229 v = f[s \u2229 f\u207b\u00b9[v]] &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s : Set \u03b1)\nvariable (v : Set \u03b2)\n\nexample : (f '' s) \u2229 v = f '' (s \u2229 f \u207b\u00b9' v) :=\nby sorry\n<\/pre>\n<h4>2.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Tenemmos que demostrar que, para todo &#92;(y&#92;),<br \/>\n&#92;[ y \u2208 f[s] \u2229 v \u2194 y \u2208 f[s \u2229 f\u207b\u00b9[v]] &#92;]<br \/>\nLo haremos demostrando las dos implicaciones.<\/p>\n<p>(\u27f9) Supongamos que &#92;(y \u2208 f[s] \u2229 v&#92;). Entonces,<br \/>\n&#92;begin{align}<br \/>\n   &amp;y \u2208 f[s] &#92;tag{1} &#92;&#92;<br \/>\n   &amp;y \u2208 v    &#92;tag{2}<br \/>\n&#92;end{align}<br \/>\nPor (1), existe un &#92;(x&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s    &#92;tag{3} &#92;&#92;<br \/>\n   &amp;f(x) = y &#92;tag{4}<br \/>\n&#92;end{align}<br \/>\nDe (2) y (4), se tiene que<br \/>\n&#92;[ f(x) \u2208 v &#92;]<br \/>\ny, por tanto,<br \/>\n&#92;[ x \u2208 f\u207b\u00b9[v] &#92;tag{5} &#92;]<br \/>\nDe (3) y (5), se tiene que<br \/>\n&#92;[ x \u2208 s \u2229 f\u207b\u00b9[v] &#92;]<br \/>\nPor tanto,<br \/>\n&#92;[ f(x) \u2208 f[s \u2229 f\u207b\u00b9[v]] &#92;]<br \/>\ny, por (4),<br \/>\n&#92;[ y \u2208 f[s \u2229 f\u207b\u00b9[v]] &#92;]<\/p>\n<p>(\u27f8) Supongamos que &#92;(y \u2208 f[s \u2229 f\u207b\u00b9[v]]&#92;). Entonces, existe un &#92;(x&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s \u2229 f\u207b\u00b9[v] &#92;tag{6} &#92;&#92;<br \/>\n   &amp;f(x) = y       &#92;tag{7}<br \/>\n&#92;end{align}<br \/>\nPor (6), se tiene que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 s       &#92;tag{8} &#92;&#92;<br \/>\n   &amp;x \u2208 f\u207b\u00b9[v]  &#92;tag{9}<br \/>\n&#92;end{align}<br \/>\nPor (8), se tiene que<br \/>\n&#92;[ f(x) \u2208 f[s] &#92;]<br \/>\ny, por (7),<br \/>\n&#92;[ y \u2208 f[s] &#92;tag{10} &#92;]<br \/>\nPor (9),<br \/>\n&#92;[ f(x) \u2208 v &#92;]<br \/>\ny, por (7),<br \/>\n&#92;[ y \u2208 v &#92;tag{11} &#92;]<br \/>\nPor (10) y (11),<br \/>\n&#92;[ y \u2208 f[s] \u2229 v &#92;]<\/p>\n<h4>2.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s : Set \u03b1)\nvariable (v : Set \u03b2)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 v = f '' (s \u2229 f \u207b\u00b9' v) :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' s \u2229 v \u2194 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n  constructor\n  . -- \u22a2 y \u2208 f '' s \u2229 v \u2192 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    intro hy\n    -- hy : y \u2208 f '' s \u2229 v\n    -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    cases' hy with hyfs yv\n    -- hyfs : y \u2208 f '' s\n    -- yv : y \u2208 v\n    cases' hyfs with x hx\n    -- x : \u03b1\n    -- hx : x \u2208 s \u2227 f x = y\n    cases' hx with xs fxy\n    -- xs : x \u2208 s\n    -- fxy : f x = y\n    have h1 : f x \u2208 v := by rwa [\u2190fxy] at yv\n    have h3 : x \u2208 s \u2229 f \u207b\u00b9' v := mem_inter xs h1\n    have h4 : f x \u2208 f '' (s \u2229 f \u207b\u00b9' v) := mem_image_of_mem f h3\n    show y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    rwa [fxy] at h4\n  . -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' v) \u2192 y \u2208 f '' s \u2229 v\n    intro hy\n    -- hy : y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    -- \u22a2 y \u2208 f '' s \u2229 v\n    cases' hy with x hx\n    -- x : \u03b1\n    -- hx : x \u2208 s \u2229 f \u207b\u00b9' v \u2227 f x = y\n    cases' hx with hx1 fxy\n    -- hx1 : x \u2208 s \u2229 f \u207b\u00b9' v\n    -- fxy : f x = y\n    cases' hx1 with xs xfv\n    -- xs : x \u2208 s\n    -- xfv : x \u2208 f \u207b\u00b9' v\n    have h5 : f x \u2208 f '' s := mem_image_of_mem f xs\n    have h6 : y \u2208 f '' s := by rwa [fxy] at h5\n    have h7 : f x \u2208 v := mem_preimage.mp xfv\n    have h8 : y \u2208 v := by rwa [fxy] at h7\n    show y \u2208 f '' s \u2229 v\n    exact mem_inter h6 h8\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 v = f '' (s \u2229 f \u207b\u00b9' v) :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' s \u2229 v \u2194 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n  constructor\n  . -- \u22a2 y \u2208 f '' s \u2229 v \u2192 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    intro hy\n    -- hy : y \u2208 f '' s \u2229 v\n    -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    cases' hy with hyfs yv\n    -- hyfs : y \u2208 f '' s\n    -- yv : y \u2208 v\n    cases' hyfs with x hx\n    -- x : \u03b1\n    -- hx : x \u2208 s \u2227 f x = y\n    cases' hx with xs fxy\n    -- xs : x \u2208 s\n    -- fxy : f x = y\n    use x\n    -- \u22a2 x \u2208 s \u2229 f \u207b\u00b9' v \u2227 f x = y\n    constructor\n    . -- \u22a2 x \u2208 s \u2229 f \u207b\u00b9' v\n      constructor\n      . -- \u22a2 x \u2208 s\n        exact xs\n      . -- \u22a2 x \u2208 f \u207b\u00b9' v\n        rw [mem_preimage]\n        -- \u22a2 f x \u2208 v\n        rw [fxy]\n        -- \u22a2 y \u2208 v\n        exact yv\n    . -- \u22a2 f x = y\n      exact fxy\n  . -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' v) \u2192 y \u2208 f '' s \u2229 v\n    intro hy\n    -- hy : y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    -- \u22a2 y \u2208 f '' s \u2229 v\n    cases' hy with x hx\n    -- x : \u03b1\n    -- hx : x \u2208 s \u2229 f \u207b\u00b9' v \u2227 f x = y\n    constructor\n    . -- \u22a2 y \u2208 f '' s\n      use x\n      -- \u22a2 x \u2208 s \u2227 f x = y\n      constructor\n      . -- \u22a2 x \u2208 s\n        exact hx.1.1\n      . -- \u22a2 f x = y\n        exact hx.2\n    . -- \u22a2 y \u2208 v\n      cases' hx with hx1 fxy\n      -- hx1 : x \u2208 s \u2229 f \u207b\u00b9' v\n      -- fxy : f x = y\n      rw [\u2190fxy]\n      -- \u22a2 f x \u2208 v\n      rw [\u2190mem_preimage]\n      -- \u22a2 x \u2208 f \u207b\u00b9' v\n      exact hx1.2\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 v = f '' (s \u2229 f \u207b\u00b9' v) :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' s \u2229 v \u2194 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n  constructor\n  . -- \u22a2 y \u2208 f '' s \u2229 v \u2192 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    rintro \u27e8\u27e8x, xs, fxy\u27e9, yv\u27e9\n    -- yv : y \u2208 v\n    -- x : \u03b1\n    -- xs : x \u2208 s\n    -- fxy : f x = y\n    -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    use x\n    -- \u22a2 x \u2208 s \u2229 f \u207b\u00b9' v \u2227 f x = y\n    constructor\n    . -- \u22a2 x \u2208 s \u2229 f \u207b\u00b9' v\n      constructor\n      . -- \u22a2 x \u2208 s\n        exact xs\n      . -- \u22a2 x \u2208 f \u207b\u00b9' v\n        rw [mem_preimage]\n        -- \u22a2 f x \u2208 v\n        rw [fxy]\n        -- \u22a2 y \u2208 v\n        exact yv\n    . exact fxy\n  . rintro \u27e8x, \u27e8xs, xv\u27e9, fxy\u27e9\n    -- x : \u03b1\n    -- fxy : f x = y\n    -- xs : x \u2208 s\n    -- xv : x \u2208 f \u207b\u00b9' v\n    -- \u22a2 y \u2208 f '' s \u2229 v\n    constructor\n    . -- \u22a2 y \u2208 f '' s\n      use x, xs\n      -- \u22a2 f x = y\n      exact fxy\n    . -- \u22a2 y \u2208 v\n      rw [\u2190fxy]\n      -- \u22a2 f x \u2208 v\n      rw [\u2190mem_preimage]\n      -- \u22a2 x \u2208 f \u207b\u00b9' v\n      exact xv\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 v = f '' (s \u2229 f \u207b\u00b9' v) :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' s \u2229 v \u2194 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n  constructor\n  . -- \u22a2 y \u2208 f '' s \u2229 v \u2192 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    rintro \u27e8\u27e8x, xs, fxy\u27e9, yv\u27e9\n    -- yv : y \u2208 v\n    -- x : \u03b1\n    -- xs : x \u2208 s\n    -- fxy : f x = y\n    -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' v)\n    aesop\n  . -- \u22a2 y \u2208 f '' (s \u2229 f \u207b\u00b9' v) \u2192 y \u2208 f '' s \u2229 v\n    rintro \u27e8x, \u27e8xs, xv\u27e9, fxy\u27e9\n    -- x : \u03b1\n    -- fxy : f x = y\n    -- xs : x \u2208 s\n    -- xv : x \u2208 f \u207b\u00b9' v\n    -- \u22a2 y \u2208 f '' s \u2229 v\n    aesop\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 v = f '' (s \u2229 f \u207b\u00b9' v) :=\nby ext ; constructor <;> aesop\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (f '' s) \u2229 v = f '' (s \u2229 f \u207b\u00b9' v) :=\n(image_inter_preimage f s v).symm\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (a b : Set \u03b1)\n-- #check (image_inter_preimage f s v : f '' (s \u2229 f \u207b\u00b9' v) = f '' s \u2229 v)\n-- #check (mem_image_of_mem  f : x \u2208 a \u2192 f x \u2208 f '' a)\n-- #check (mem_inter : x \u2208 a \u2192 x \u2208 b \u2192 x \u2208 a \u2229 b)\n-- #check (mem_preimage : x \u2208 f \u207b\u00b9' v \u2194 f x \u2208 v)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Interseccion_con_la_imagen_inversa.lean\">Lean 4 Web<\/a>.<\/p>\n<h4>2.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Interseccion_con_la_imagen_inversa\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"(f ` s) \u2229 v = f ` (s \u2229 f -` v)\"\nproof (rule equalityI)\n  show \"(f ` s) \u2229 v \u2286 f ` (s \u2229 f -` v)\"\n  proof (rule subsetI)\n    fix y\n    assume \"y \u2208 (f ` s) \u2229 v\"\n    then show \"y \u2208 f ` (s \u2229 f -` v)\"\n    proof (rule IntE)\n      assume \"y \u2208 v\"\n      assume \"y \u2208 f ` s\"\n      then show \"y \u2208 f ` (s \u2229 f -` v)\"\n      proof (rule imageE)\n        fix x\n        assume \"x \u2208 s\"\n        assume \"y = f x\"\n        then have \"f x \u2208 v\"\n          using \u2039y \u2208 v\u203a by (rule subst)\n        then have \"x \u2208 f -` v\"\n          by (rule vimageI2)\n        with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 f -` v\"\n          by (rule IntI)\n        then have \"f x \u2208 f ` (s \u2229 f -` v)\"\n          by (rule imageI)\n        with \u2039y = f x\u203a show \"y \u2208 f ` (s \u2229 f -` v)\"\n          by (rule ssubst)\n      qed\n    qed\n  qed\nnext\n  show \"f ` (s \u2229 f -` v) \u2286 (f ` s) \u2229 v\"\n  proof (rule subsetI)\n    fix y\n    assume \"y \u2208 f ` (s \u2229 f -` v)\"\n    then show \"y \u2208 (f ` s) \u2229 v\"\n    proof (rule imageE)\n      fix x\n      assume \"y = f x\"\n      assume hx : \"x \u2208 s \u2229 f -` v\"\n      have \"y \u2208 f ` s\"\n      proof -\n        have \"x \u2208 s\"\n          using hx by (rule IntD1)\n        then have \"f x \u2208 f ` s\"\n          by (rule imageI)\n        with \u2039y = f x\u203a show \"y \u2208 f ` s\"\n          by (rule ssubst)\n      qed\n      moreover\n      have \"y \u2208 v\"\n      proof -\n        have \"x \u2208 f -` v\"\n          using hx by (rule IntD2)\n        then have \"f x \u2208 v\"\n          by (rule vimageD)\n        with \u2039y = f x\u203a show \"y \u2208 v\"\n          by (rule ssubst)\n      qed\n      ultimately show \"y \u2208 (f ` s) \u2229 v\"\n        by (rule IntI)\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"(f ` s) \u2229 v = f ` (s \u2229 f -` v)\"\nproof\n  show \"(f ` s) \u2229 v \u2286 f ` (s \u2229 f -` v)\"\n  proof\n    fix y\n    assume \"y \u2208 (f ` s) \u2229 v\"\n    then show \"y \u2208 f ` (s \u2229 f -` v)\"\n    proof\n      assume \"y \u2208 v\"\n      assume \"y \u2208 f ` s\"\n      then show \"y \u2208 f ` (s \u2229 f -` v)\"\n      proof\n        fix x\n        assume \"x \u2208 s\"\n        assume \"y = f x\"\n        then have \"f x \u2208 v\" using \u2039y \u2208 v\u203a by simp\n        then have \"x \u2208 f -` v\" by simp\n        with \u2039x \u2208 s\u203a have \"x \u2208 s \u2229 f -` v\" by simp\n        then have \"f x \u2208 f ` (s \u2229 f -` v)\" by simp\n        with \u2039y = f x\u203a show \"y \u2208 f ` (s \u2229 f -` v)\" by simp\n      qed\n    qed\n  qed\nnext\n  show \"f ` (s \u2229 f -` v) \u2286 (f ` s) \u2229 v\"\n  proof\n    fix y\n    assume \"y \u2208 f ` (s \u2229 f -` v)\"\n    then show \"y \u2208 (f ` s) \u2229 v\"\n    proof\n      fix x\n      assume \"y = f x\"\n      assume hx : \"x \u2208 s \u2229 f -` v\"\n      have \"y \u2208 f ` s\"\n      proof -\n        have \"x \u2208 s\" using hx by simp\n        then have \"f x \u2208 f ` s\" by simp\n        with \u2039y = f x\u203a show \"y \u2208 f ` s\" by simp\n      qed\n      moreover\n      have \"y \u2208 v\"\n      proof -\n        have \"x \u2208 f -` v\" using hx by simp\n        then have \"f x \u2208 v\" by simp\n        with \u2039y = f x\u203a show \"y \u2208 v\" by simp\n      qed\n      ultimately show \"y \u2208 (f ` s) \u2229 v\" by simp\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"(f ` s) \u2229 v = f ` (s \u2229 f -` v)\"\nproof\n  show \"(f ` s) \u2229 v \u2286 f ` (s \u2229 f -` v)\"\n  proof\n    fix y\n    assume \"y \u2208 (f ` s) \u2229 v\"\n    then show \"y \u2208 f ` (s \u2229 f -` v)\"\n    proof\n      assume \"y \u2208 v\"\n      assume \"y \u2208 f ` s\"\n      then show \"y \u2208 f ` (s \u2229 f -` v)\"\n      proof\n        fix x\n        assume \"x \u2208 s\"\n        assume \"y = f x\"\n        then show \"y \u2208 f ` (s \u2229 f -` v)\"\n          using \u2039x \u2208 s\u203a \u2039y \u2208 v\u203a by simp\n      qed\n    qed\n  qed\nnext\n  show \"f ` (s \u2229 f -` v) \u2286 (f ` s) \u2229 v\"\n  proof\n    fix y\n    assume \"y \u2208 f ` (s \u2229 f -` v)\"\n    then show \"y \u2208 (f ` s) \u2229 v\"\n    proof\n      fix x\n      assume \"y = f x\"\n      assume hx : \"x \u2208 s \u2229 f -` v\"\n      then have \"y \u2208 f ` s\" using \u2039y = f x\u203a by simp\n      moreover\n      have \"y \u2208 v\" using hx \u2039y = f x\u203a by simp\n      ultimately show \"y \u2208 (f ` s) \u2229 v\" by simp\n    qed\n  qed\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"(f ` s) \u2229 v = f ` (s \u2229 f -` v)\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej3\"><\/a><\/p>\n<h3>3. Uni\u00f3n con la imagen inversa<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ s \u222a f\u207b\u00b9[v] \u2286 f\u207b\u00b9[f[s] \u222a v] &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s : Set \u03b1)\nvariable (v : Set \u03b2)\n\nexample : s \u222a f \u207b\u00b9' v \u2286 f \u207b\u00b9' (f '' s \u222a v) :=\nby sorry\n<\/pre>\n<h4>3.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Sea &#92;(x \u2208 s \u222a f\u207b\u00b9[v]&#92;). Entonces, se pueden dar dos casos.<\/p>\n<p>Caso 1: Supongamos que &#92;(x \u2208 s&#92;). Entonces, se tiene<br \/>\n&#92;begin{align}<br \/>\n   &amp;f(x) \u2208 f[s]       &#92;&#92;<br \/>\n   &amp;f(x) \u2208 f[s] \u222a v   &#92;&#92;<br \/>\n   &amp;x \u2208 f\u207b\u00b9[f[s] \u222a v]<br \/>\n&#92;end{align}<\/p>\n<p>Caso 2: Supongamos que x \u2208 f\u207b\u00b9[v]. Entonces, se tiene<br \/>\n&#92;begin{align}<br \/>\n   &amp;f(x) \u2208 v           &#92;&#92;<br \/>\n   &amp;f(x) \u2208 f[s] \u222a v    &#92;&#92;<br \/>\n   &amp;x \u2208 f\u207b\u00b9[f[s] \u222a v]<br \/>\n&#92;end{align}<\/p>\n<h4>3.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Function\n\nopen Set\n\nvariable {\u03b1 \u03b2 : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (s : Set \u03b1)\nvariable (v : Set \u03b2)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a f \u207b\u00b9' v \u2286 f \u207b\u00b9' (f '' s \u222a v) :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 s \u222a f \u207b\u00b9' v\n  -- \u22a2 x \u2208 f \u207b\u00b9' (f '' s \u222a v)\n  cases' hx with xs xv\n  . -- xs : x \u2208 s\n    have h1 : f x \u2208 f '' s := mem_image_of_mem f xs\n    have h2 : f x \u2208 f '' s \u222a v := mem_union_left v h1\n    show x \u2208 f \u207b\u00b9' (f '' s \u222a v)\n    exact mem_preimage.mpr h2\n  . -- xv : x \u2208 f \u207b\u00b9' v\n    have h3 : f x \u2208 v := mem_preimage.mp xv\n    have h4 : f x \u2208 f '' s \u222a v := mem_union_right (f '' s) h3\n    show x \u2208 f \u207b\u00b9' (f '' s \u222a v)\n    exact mem_preimage.mpr h4\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a f \u207b\u00b9' v \u2286 f \u207b\u00b9' (f '' s \u222a v) :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 s \u222a f \u207b\u00b9' v\n  -- \u22a2 x \u2208 f \u207b\u00b9' (f '' s \u222a v)\n  rw [mem_preimage]\n  -- \u22a2 f x \u2208 f '' s \u222a v\n  cases' hx with xs xv\n  . -- xs : x \u2208 s\n    apply mem_union_left\n    -- \u22a2 f x \u2208 f '' s\n    apply mem_image_of_mem\n    -- \u22a2 x \u2208 s\n    exact xs\n  . -- xv : x \u2208 f \u207b\u00b9' v\n    apply mem_union_right\n    -- \u22a2 f x \u2208 v\n    rw [\u2190mem_preimage]\n    -- \u22a2 x \u2208 f \u207b\u00b9' v\n    exact xv\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a f \u207b\u00b9' v \u2286 f \u207b\u00b9' (f '' s \u222a v) :=\nby\n  intros x hx\n  -- x : \u03b1\n  -- hx : x \u2208 s \u222a f \u207b\u00b9' v\n  -- \u22a2 x \u2208 f \u207b\u00b9' (f '' s \u222a v)\n  cases' hx with xs xv\n  . -- xs : x \u2208 s\n    rw [mem_preimage]\n    -- \u22a2 f x \u2208 f '' s \u222a v\n    apply mem_union_left\n    -- \u22a2 f x \u2208 f '' s\n    apply mem_image_of_mem\n    -- \u22a2 x \u2208 s\n    exact xs\n  . -- \u22a2 x \u2208 f \u207b\u00b9' (f '' s \u222a v)\n    rw [mem_preimage]\n    -- \u22a2 f x \u2208 f '' s \u222a v\n    apply mem_union_right\n    -- \u22a2 f x \u2208 v\n    exact xv\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a f \u207b\u00b9' v \u2286 f \u207b\u00b9' (f '' s \u222a v) :=\nby\n  rintro x (xs | xv)\n  -- x : \u03b1\n  -- \u22a2 x \u2208 f \u207b\u00b9' (f '' s \u222a v)\n  . -- xs : x \u2208 s\n    left\n    -- \u22a2 f x \u2208 f '' s\n    exact mem_image_of_mem f xs\n  . -- xv : x \u2208 f \u207b\u00b9' v\n    right\n    -- \u22a2 f x \u2208 v\n    exact xv\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a f \u207b\u00b9' v \u2286 f \u207b\u00b9' (f '' s \u222a v) :=\nby\n  rintro x (xs | xv)\n  -- x : \u03b1\n  -- \u22a2 x \u2208 f \u207b\u00b9' (f '' s \u222a v)\n  . -- xs : x \u2208 s\n    exact Or.inl (mem_image_of_mem f xs)\n  . -- xv : x \u2208 f \u207b\u00b9' v\n    exact Or.inr xv\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a f \u207b\u00b9' v \u2286 f \u207b\u00b9' (f '' s \u222a v) :=\nby\n  intros x h\n  -- x : \u03b1\n  -- h : x \u2208 s \u222a f \u207b\u00b9' v\n  -- \u22a2 x \u2208 f \u207b\u00b9' (f '' s \u222a v)\n  exact Or.elim h (fun xs \u21a6 Or.inl (mem_image_of_mem f xs)) Or.inr\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a f \u207b\u00b9' v \u2286 f \u207b\u00b9' (f '' s \u222a v) :=\nfun _ h \u21a6 Or.elim h (fun xs \u21a6 Or.inl (mem_image_of_mem f xs)) Or.inr\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a f \u207b\u00b9' v \u2286 f \u207b\u00b9' (f '' s \u222a v) :=\nunion_preimage_subset s v f\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (t : Set \u03b1)\n-- variable (a b c : Prop)\n-- #check (Or.elim : a \u2228 b \u2192 (a \u2192 c) \u2192 (b \u2192 c) \u2192 c)\n-- #check (Or.inl : a \u2192 a \u2228 b)\n-- #check (Or.inr : b \u2192 a \u2228 b)\n-- #check (mem_image_of_mem f : x  \u2208 s \u2192 f x \u2208 f '' s)\n-- #check (mem_preimage : x \u2208 f \u207b\u00b9' v \u2194 f x \u2208 v)\n-- #check (mem_union_left t : x \u2208 s \u2192 x \u2208 s \u222a t)\n-- #check (mem_union_right s : x \u2208 t \u2192 x \u2208 s \u222a t)\n-- #check (union_preimage_subset s v f : s \u222a f \u207b\u00b9' v \u2286 f \u207b\u00b9' (f '' s \u222a v))\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Union_con_la_imagen_inversa.lean\">Lean 4 Web<\/a>.<\/p>\n<h4>3.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Union_con_la_imagen_inversa\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"s \u222a f -` v \u2286 f -` (f ` s \u222a v)\"\nproof (rule subsetI)\n  fix x\n  assume \"x \u2208 s \u222a f -` v\"\n  then have \"f x \u2208 f ` s \u222a v\"\n  proof (rule UnE)\n    assume \"x \u2208 s\"\n    then have \"f x \u2208 f ` s\"\n      by (rule imageI)\n    then show \"f x \u2208 f ` s \u222a v\"\n      by (rule UnI1)\n  next\n    assume \"x \u2208 f -` v\"\n    then have \"f x \u2208 v\"\n      by (rule vimageD)\n    then show \"f x \u2208 f ` s \u222a v\"\n      by (rule UnI2)\n  qed\n  then show \"x \u2208 f -` (f ` s \u222a v)\"\n    by (rule vimageI2)\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"s \u222a f -` v \u2286 f -` (f ` s \u222a v)\"\nproof\n  fix x\n  assume \"x \u2208 s \u222a f -` v\"\n  then have \"f x \u2208 f ` s \u222a v\"\n  proof\n    assume \"x \u2208 s\"\n    then have \"f x \u2208 f ` s\" by simp\n    then show \"f x \u2208 f ` s \u222a v\" by simp\n  next\n    assume \"x \u2208 f -` v\"\n    then have \"f x \u2208 v\" by simp\n    then show \"f x \u2208 f ` s \u222a v\" by simp\n  qed\n  then show \"x \u2208 f -` (f ` s \u222a v)\" by simp\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma \"s \u222a f -` v \u2286 f -` (f ` s \u222a v)\"\nproof\n  fix x\n  assume \"x \u2208 s \u222a f -` v\"\n  then have \"f x \u2208 f ` s \u222a v\"\n  proof\n    assume \"x \u2208 s\"\n    then show \"f x \u2208 f ` s \u222a v\" by simp\n  next\n    assume \"x \u2208 f -` v\"\n    then show \"f x \u2208 f ` s \u222a v\" by simp\n  qed\n  then show \"x \u2208 f -` (f ` s \u222a v)\" by simp\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"s \u222a f -` v \u2286 f -` (f ` s \u222a v)\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej4\"><\/a><\/p>\n<h3>4. Imagen de la uni\u00f3n general<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ f[\u22c3\u1d62A\u1d62] = \u22c3\u1d62f[A\u1d62] &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 I : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (A : \u2115 \u2192 Set \u03b1)\n\nexample : f '' (\u22c3 i, A i) = \u22c3 i, f '' A i :=\nby sorry\n<\/pre>\n<h4>4.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Tenemos que demostrar que, para todo &#92;(y&#92;),<br \/>\n&#92;[ y \u2208 f[\u22c3\u1d62A\u1d62] \u2194 y \u2208 \u22c3\u1d62f[A\u1d62] &#92;]<br \/>\nLo haremos demostrando las dos implicaciones.<\/p>\n<p>(\u27f9) Supongamos que &#92;(y \u2208 f[\u22c3\u1d62A\u1d62]&#92;). Entonces, existe un &#92;(x&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 \u22c3\u1d62A\u1d62 &#92;tag{1} &#92;&#92;<br \/>\n   &amp;f(x) = y &#92;tag{2}<br \/>\n&#92;end{align}<br \/>\nPor (1), existe un i tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;i \u2208 \u2115  &#92;tag{3} &#92;&#92;<br \/>\n   &amp;x \u2208 A\u1d62 &#92;tag{4}<br \/>\n&#92;end{align}<br \/>\nPor (4),<br \/>\n&#92;[ f(x) \u2208 f[A\u1d62] &#92;]<br \/>\nPor (3),<br \/>\n&#92;[ f(x) \u2208 \u22c3\u1d62f[A\u1d62] &#92;]<br \/>\ny, por (2),<br \/>\n&#92;[ y \u2208 \u22c3\u1d62f[A\u1d62] &#92;]<\/p>\n<p>(\u27f8) Supongamos que &#92;(y \u2208 \u22c3\u1d62f[A\u1d62]&#92;). Entonces, existe un &#92;(i&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;i \u2208 \u2115     &#92;tag{5} &#92;&#92;<br \/>\n   &amp;y \u2208 f[A\u1d62] &#92;tag{6}<br \/>\n&#92;end{align}<br \/>\nPor (6), existe un &#92;(x&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 A\u1d62   &#92;tag{7} &#92;&#92;<br \/>\n   &amp;f(x) = y &#92;tag{8}<br \/>\n&#92;end{align}<br \/>\nPor (5) y (7),<br \/>\n&#92;[ x \u2208 \u22c3\u1d62A\u1d62 &#92;]<br \/>\nLuego,<br \/>\n&#92;[ f(x) \u2208 f[\u22c3\u1d62A\u1d62] &#92;]<br \/>\ny, por (8),<br \/>\n&#92;[ y \u2208 f[\u22c3\u1d62A\u1d62] &#92;]<\/p>\n<h4>4.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 I : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (A : \u2115 \u2192 Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (\u22c3 i, A i) = \u22c3 i, f '' A i :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' \u22c3 (i : \u2115), A i \u2194 y \u2208 \u22c3 (i : \u2115), f '' A i\n  constructor\n  . -- \u22a2 y \u2208 f '' \u22c3 (i : \u2115), A i \u2192 y \u2208 \u22c3 (i : \u2115), f '' A i\n    intro hy\n    -- hy : y \u2208 f '' \u22c3 (i : \u2115), A i\n    -- \u22a2 y \u2208 \u22c3 (i : \u2115), f '' A i\n    have h1 : \u2203 x, x \u2208 \u22c3 i, A i \u2227 f x = y := (mem_image f (\u22c3 i, A i) y).mp hy\n    obtain \u27e8x, hx : x \u2208 \u22c3 i, A i \u2227 f x = y\u27e9 := h1\n    have xUA : x \u2208 \u22c3 i, A i := hx.1\n    have fxy : f x = y := hx.2\n    have xUA : \u2203 i, x \u2208 A i := mem_iUnion.mp xUA\n    obtain \u27e8i, xAi : x \u2208 A i\u27e9 := xUA\n    have h2 : f x \u2208 f '' A i := mem_image_of_mem f xAi\n    have h3 : f x \u2208 \u22c3 i, f '' A i := mem_iUnion_of_mem i h2\n    show y \u2208 \u22c3 i, f '' A i\n    rwa [fxy] at h3\n  . -- \u22a2 y \u2208 \u22c3 (i : \u2115), f '' A i \u2192 y \u2208 f '' \u22c3 (i : \u2115), A i\n    intro hy\n    -- hy : y \u2208 \u22c3 (i : \u2115), f '' A i\n    -- \u22a2 y \u2208 f '' \u22c3 (i : \u2115), A i\n    have h4 : \u2203 i, y \u2208 f '' A i := mem_iUnion.mp hy\n    obtain \u27e8i, h5 : y \u2208 f '' A i\u27e9 := h4\n    have h6 : \u2203 x, x \u2208 A i \u2227 f x = y := (mem_image f (A i) y).mp h5\n    obtain \u27e8x, h7 : x \u2208 A i \u2227 f x = y\u27e9 := h6\n    have h8 : x \u2208 A i := h7.1\n    have h9 : x \u2208 \u22c3 i, A i := mem_iUnion_of_mem i h8\n    have h10 : f x \u2208 f '' (\u22c3 i, A i) := mem_image_of_mem f h9\n    show y \u2208 f '' (\u22c3 i, A i)\n    rwa [h7.2] at h10\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (\u22c3 i, A i) = \u22c3 i, f '' A i :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' \u22c3 (i : \u2115), A i \u2194 y \u2208 \u22c3 (i : \u2115), f '' A i\n  constructor\n  . -- \u22a2 y \u2208 f '' \u22c3 (i : \u2115), A i \u2192 y \u2208 \u22c3 (i : \u2115), f '' A i\n    intro hy\n    -- hy : y \u2208 f '' \u22c3 (i : \u2115), A i\n    -- \u22a2 y \u2208 \u22c3 (i : \u2115), f '' A i\n    rw [mem_image] at hy\n    -- hy : \u2203 x, x \u2208 \u22c3 (i : \u2115), A i \u2227 f x = y\n    cases' hy with x hx\n    -- x : \u03b1\n    -- hx : x \u2208 \u22c3 (i : \u2115), A i \u2227 f x = y\n    cases' hx with xUA fxy\n    -- xUA : x \u2208 \u22c3 (i : \u2115), A i\n    -- fxy : f x = y\n    rw [mem_iUnion] at xUA\n    -- xUA : \u2203 i, x \u2208 A i\n    cases' xUA with i xAi\n    -- i : \u2115\n    -- xAi : x \u2208 A i\n    rw [mem_iUnion]\n    -- \u22a2 \u2203 i, y \u2208 f '' A i\n    use i\n    -- \u22a2 y \u2208 f '' A i\n    rw [\u2190fxy]\n    -- \u22a2 f x \u2208 f '' A i\n    apply mem_image_of_mem\n    -- \u22a2 x \u2208 A i\n    exact xAi\n  . -- \u22a2 y \u2208 \u22c3 (i : \u2115), f '' A i \u2192 y \u2208 f '' \u22c3 (i : \u2115), A i\n    intro hy\n    -- hy : y \u2208 \u22c3 (i : \u2115), f '' A i\n    -- \u22a2 y \u2208 f '' \u22c3 (i : \u2115), A i\n    rw [mem_iUnion] at hy\n    -- hy : \u2203 i, y \u2208 f '' A i\n    cases' hy with i yAi\n    -- i : \u2115\n    -- yAi : y \u2208 f '' A i\n    cases' yAi with x hx\n    -- x : \u03b1\n    -- hx : x \u2208 A i \u2227 f x = y\n    cases' hx with xAi fxy\n    -- xAi : x \u2208 A i\n    -- fxy : f x = y\n    rw [\u2190fxy]\n    -- \u22a2 f x \u2208 f '' \u22c3 (i : \u2115), A i\n    apply mem_image_of_mem\n    -- \u22a2 x \u2208 \u22c3 (i : \u2115), A i\n    rw [mem_iUnion]\n    -- \u22a2 \u2203 i, x \u2208 A i\n    use i\n    -- \u22a2 x \u2208 A i\n    exact xAi\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (\u22c3 i, A i) = \u22c3 i, f '' A i :=\nby\n  ext y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' \u22c3 (i : \u2115), A i \u2194 y \u2208 \u22c3 (i : \u2115), f '' A i\n  simp\n  -- \u22a2 (\u2203 x, (\u2203 i, x \u2208 A i) \u2227 f x = y) \u2194 \u2203 i x, x \u2208 A i \u2227 f x = y\n  constructor\n  . -- \u22a2 (\u2203 x, (\u2203 i, x \u2208 A i) \u2227 f x = y) \u2192 \u2203 i x, x \u2208 A i \u2227 f x = y\n    rintro \u27e8x, \u27e8i, xAi\u27e9, fxy\u27e9\n    -- x : \u03b1\n    -- fxy : f x = y\n    -- i : \u2115\n    -- xAi : x \u2208 A i\n    -- \u22a2 \u2203 i x, x \u2208 A i \u2227 f x = y\n    use i, x, xAi\n    -- \u22a2 f x = y\n    exact fxy\n  . -- \u22a2 (\u2203 i x, x \u2208 A i \u2227 f x = y) \u2192 \u2203 x, (\u2203 i, x \u2208 A i) \u2227 f x = y\n    rintro \u27e8i, x, xAi, fxy\u27e9\n    -- i : \u2115\n    -- x : \u03b1\n    -- xAi : x \u2208 A i\n    -- fxy : f x = y\n    -- \u22a2 \u2203 x, (\u2203 i, x \u2208 A i) \u2227 f x = y\n    exact \u27e8x, \u27e8i, xAi\u27e9, fxy\u27e9\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (\u22c3 i, A i) = \u22c3 i, f '' A i :=\nimage_iUnion\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (y : \u03b2)\n-- variable (s : Set \u03b1)\n-- variable (i : \u2115)\n-- #check (image_iUnion : f '' \u22c3 i, A i = \u22c3 i, f '' A i)\n-- #check (mem_iUnion : x \u2208 \u22c3 i, A i \u2194 \u2203 i, x \u2208 A i)\n-- #check (mem_iUnion_of_mem i : x \u2208 A i \u2192 x \u2208 \u22c3 i, A i)\n-- #check (mem_image f s y : (y \u2208 f '' s \u2194 \u2203 x, x \u2208 s \u2227 f x = y))\n-- #check (mem_image_of_mem f : x  \u2208 s \u2192 f x \u2208 f '' s)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Imagen_de_la_union_general.lean\">Lean 4 Web<\/a>.<\/p>\n<h4>4.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Imagen_de_la_union_general\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. f ` A i)\"\nproof (rule equalityI)\n  show \"f ` (\u22c3 i \u2208 I. A i) \u2286 (\u22c3 i \u2208 I. f ` A i)\"\n  proof (rule subsetI)\n    fix y\n    assume \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\n    then show \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\"\n    proof (rule imageE)\n      fix x\n      assume \"y = f x\"\n      assume \"x \u2208 (\u22c3 i \u2208 I. A i)\"\n      then have \"f x \u2208 (\u22c3 i \u2208 I. f ` A i)\"\n      proof (rule UN_E)\n        fix i\n        assume \"i \u2208 I\"\n        assume \"x \u2208 A i\"\n        then have \"f x \u2208 f ` A i\"\n          by (rule imageI)\n        with \u2039i \u2208 I\u203a show \"f x \u2208 (\u22c3 i \u2208 I. f ` A i)\"\n          by (rule UN_I)\n      qed\n      with \u2039y = f x\u203a show \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\"\n        by (rule ssubst)\n    qed\n  qed\nnext\n  show \"(\u22c3 i \u2208 I. f ` A i) \u2286 f ` (\u22c3 i \u2208 I. A i)\"\n  proof (rule subsetI)\n    fix y\n    assume \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\"\n    then show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\n    proof (rule UN_E)\n      fix i\n      assume \"i \u2208 I\"\n      assume \"y \u2208 f ` A i\"\n      then show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\n      proof (rule imageE)\n        fix x\n        assume \"y = f x\"\n        assume \"x \u2208 A i\"\n        with \u2039i \u2208 I\u203a have \"x \u2208 (\u22c3 i \u2208 I. A i)\"\n          by (rule UN_I)\n        then have \"f x \u2208 f ` (\u22c3 i \u2208 I. A i)\"\n          by (rule imageI)\n        with \u2039y = f x\u203a show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\n          by (rule ssubst)\n      qed\n    qed\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. f ` A i)\"\nproof\n  show \"f ` (\u22c3 i \u2208 I. A i) \u2286 (\u22c3 i \u2208 I. f ` A i)\"\n  proof\n    fix y\n    assume \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\n    then show \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\"\n    proof\n      fix x\n      assume \"y = f x\"\n      assume \"x \u2208 (\u22c3 i \u2208 I. A i)\"\n      then have \"f x \u2208 (\u22c3 i \u2208 I. f ` A i)\"\n      proof\n        fix i\n        assume \"i \u2208 I\"\n        assume \"x \u2208 A i\"\n        then have \"f x \u2208 f ` A i\" by simp\n        with \u2039i \u2208 I\u203a show \"f x \u2208 (\u22c3 i \u2208 I. f ` A i)\" by (rule UN_I)\n      qed\n      with \u2039y = f x\u203a show \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\" by simp\n    qed\n  qed\nnext\n  show \"(\u22c3 i \u2208 I. f ` A i) \u2286 f ` (\u22c3 i \u2208 I. A i)\"\n  proof\n    fix y\n    assume \"y \u2208 (\u22c3 i \u2208 I. f ` A i)\"\n    then show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\n    proof\n      fix i\n      assume \"i \u2208 I\"\n      assume \"y \u2208 f ` A i\"\n      then show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\"\n      proof\n        fix x\n        assume \"y = f x\"\n        assume \"x \u2208 A i\"\n        with \u2039i \u2208 I\u203a have \"x \u2208 (\u22c3 i \u2208 I. A i)\" by (rule UN_I)\n        then have \"f x \u2208 f ` (\u22c3 i \u2208 I. A i)\" by simp\n        with \u2039y = f x\u203a show \"y \u2208 f ` (\u22c3 i \u2208 I. A i)\" by simp\n      qed\n    qed\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. f ` A i)\"\n  by (simp only: image_UN)\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (\u22c3 i \u2208 I. A i) = (\u22c3 i \u2208 I. f ` A i)\"\n  by auto\n\nend\n<\/pre>\n<p><a name=\"ej5\"><\/a><\/p>\n<h3>5. Imagen de la intersecci\u00f3n general<\/h3>\n<p>Demostrar con Lean4 que<br \/>\n&#92;[ f&#92;left[&#92;bigcap_{i \u2208 I} A_i&#92;right] \u2286 &#92;bigcap_{i \u2208 I} f[A_i] &#92;]<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 I : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (A : I \u2192 Set \u03b1)\n\nexample : f '' (\u22c2 i, A i) \u2286 \u22c2 i, f '' A i :=\nby sorry\n<\/pre>\n<h4>5.1. Demostraci\u00f3n en lenguaje natural<\/h4>\n<p>Sea &#92;(y&#92;) tal que<br \/>\n&#92;[ y \u2208 f&#92;left[&#92;bigcap_{i \u2208 I} A\u1d62&#92;right] &#92;tag{1}  &#92;]<br \/>\nTenemos que demostrar que<br \/>\n&#92;[ y \u2208 &#92;bigcap_{i \u2208 I} f[A\u1d62] &#92;]<br \/>\nPara ello, sea &#92;(i \u2208 I&#92;), tenemos que demostrar que &#92;(y \u2208 f[A\u1d62]&#92;).<\/p>\n<p>Por (1), existe un &#92;(x&#92;) tal que<br \/>\n&#92;begin{align}<br \/>\n   &amp;x \u2208 &#92;bigcap_{i \u2208 I} A\u1d62 &#92;tag{2} &#92;&#92;<br \/>\n   &amp;f(x) = y  &#92;tag{3}<br \/>\n&#92;end{align}<br \/>\nPor (2),<br \/>\n&#92;[ x \u2208 A\u1d62 &#92;]<br \/>\ny, por tanto,<br \/>\n&#92;[ f(x) \u2208 f[A\u1d62] &#92;]<br \/>\nque, junto con (3), da que<br \/>\n&#92;[ y \u2208 f[A\u1d62] &#92;]<\/p>\n<h4>5.2. Demostraciones con Lean4<\/h4>\n<pre lang=\"lean\">\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Tactic\n\nopen Set\n\nvariable {\u03b1 \u03b2 I : Type _}\nvariable (f : \u03b1 \u2192 \u03b2)\nvariable (A : I \u2192 Set \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (\u22c2 i, A i) \u2286 \u22c2 i, f '' A i :=\nby\n  intros y h\n  -- y : \u03b2\n  -- h : y \u2208 f '' \u22c2 (i : I), A i\n  -- \u22a2 y \u2208 \u22c2 (i : I), f '' A i\n  have h1 : \u2203 x, x \u2208 \u22c2 i, A i \u2227 f x = y := (mem_image f (\u22c2 i, A i) y).mp h\n  obtain \u27e8x, hx : x \u2208 \u22c2 i, A i \u2227 f x = y\u27e9 := h1\n  have h2 : x \u2208 \u22c2 i, A i := hx.1\n  have h3 : f x = y := hx.2\n  have h4 : \u2200 i, y \u2208 f '' A i := by\n    intro i\n    have h4a : x \u2208 A i := mem_iInter.mp h2 i\n    have h4b : f x \u2208 f '' A i := mem_image_of_mem f h4a\n    show y \u2208 f '' A i\n    rwa [h3] at h4b\n  show y \u2208 \u22c2 i, f '' A i\n  exact mem_iInter.mpr h4\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (\u22c2 i, A i) \u2286 \u22c2 i, f '' A i :=\nby\n  intros y h\n  -- y : \u03b2\n  -- h : y \u2208 f '' \u22c2 (i : I), A i\n  -- \u22a2 y \u2208 \u22c2 (i : I), f '' A i\n  apply mem_iInter_of_mem\n  -- \u22a2 \u2200 (i : I), y \u2208 f '' A i\n  intro i\n  -- i : I\n  -- \u22a2 y \u2208 f '' A i\n  cases' h with x hx\n  -- x : \u03b1\n  -- hx : x \u2208 \u22c2 (i : I), A i \u2227 f x = y\n  cases' hx with xIA fxy\n  -- xIA : x \u2208 \u22c2 (i : I), A i\n  -- fxy : f x = y\n  rw [\u2190fxy]\n  -- \u22a2 f x \u2208 f '' A i\n  apply mem_image_of_mem\n  -- \u22a2 x \u2208 A i\n  exact mem_iInter.mp xIA i\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (\u22c2 i, A i) \u2286 \u22c2 i, f '' A i :=\nby\n  intros y h\n  -- y : \u03b2\n  -- h : y \u2208 f '' \u22c2 (i : I), A i\n  -- \u22a2 y \u2208 \u22c2 (i : I), f '' A i\n  apply mem_iInter_of_mem\n  -- \u22a2 \u2200 (i : I), y \u2208 f '' A i\n  intro i\n  -- i : I\n  -- \u22a2 y \u2208 f '' A i\n  rcases h with \u27e8x, xIA, rfl\u27e9\n  -- x : \u03b1\n  -- xIA : x \u2208 \u22c2 (i : I), A i\n  -- \u22a2 f x \u2208 f '' A i\n  exact mem_image_of_mem f (mem_iInter.mp xIA i)\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (\u22c2 i, A i) \u2286 \u22c2 i, f '' A i :=\nby\n  intro y\n  -- y : \u03b2\n  -- \u22a2 y \u2208 f '' \u22c2 (i : I), A i \u2192 y \u2208 \u22c2 (i : I), f '' A i\n  simp\n  -- \u22a2 \u2200 (x : \u03b1), (\u2200 (i : I), x \u2208 A i) \u2192 f x = y \u2192 \u2200 (i : I), \u2203 x, x \u2208 A i \u2227 f x = y\n  intros x xIA fxy i\n  -- x : \u03b1\n  -- xIA : \u2200 (i : I), x \u2208 A i\n  -- fxy : f x = y\n  -- i : I\n  -- \u22a2 \u2203 x, x \u2208 A i \u2227 f x = y\n  use x, xIA i\n  -- \u22a2 f x = y\n  exact fxy\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : f '' (\u22c2 i, A i) \u2286 \u22c2 i, f '' A i :=\nimage_iInter_subset A f\n\n-- Lemas usados\n-- ============\n\n-- variable (x : \u03b1)\n-- variable (s : Set \u03b1)\n-- #check (image_iInter_subset A f : f '' \u22c2 i, A i \u2286 \u22c2 i, f '' A i)\n-- #check (mem_iInter : x \u2208 \u22c2 i, A i \u2194 \u2200 i, x \u2208 A i)\n-- #check (mem_iInter_of_mem : (\u2200 i, x \u2208 A i) \u2192 x \u2208 \u22c2 i, A i)\n-- #check (mem_image_of_mem f : x \u2208 s \u2192 f x \u2208 f '' s)\n<\/pre>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Imagen_de_la_interseccion_general.lean\">Lean 4 Web<\/a>.<\/p>\n<h4>5.3. Demostraciones con Isabelle\/HOL<\/h4>\n<pre lang=\"isar\">\ntheory Imagen_de_la_interseccion_general\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (\u22c2 i \u2208 I. A i) \u2286 (\u22c2 i \u2208 I. f ` A i)\"\nproof (rule subsetI)\n  fix y\n  assume \"y \u2208 f ` (\u22c2 i \u2208 I. A i)\"\n  then show \"y \u2208 (\u22c2 i \u2208 I. f ` A i)\"\n  proof (rule imageE)\n    fix x\n    assume \"y = f x\"\n    assume xIA : \"x \u2208 (\u22c2 i \u2208 I. A i)\"\n    have \"f x \u2208 (\u22c2 i \u2208 I. f ` A i)\"\n    proof (rule INT_I)\n      fix i\n      assume \"i \u2208 I\"\n      with xIA have \"x \u2208 A i\"\n        by (rule INT_D)\n      then show \"f x \u2208 f ` A i\"\n        by (rule imageI)\n    qed\n    with \u2039y = f x\u203a show \"y \u2208 (\u22c2 i \u2208 I. f ` A i)\"\n      by (rule ssubst)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (\u22c2 i \u2208 I. A i) \u2286 (\u22c2 i \u2208 I. f ` A i)\"\nproof\n  fix y\n  assume \"y \u2208 f ` (\u22c2 i \u2208 I. A i)\"\n  then show \"y \u2208 (\u22c2 i \u2208 I. f ` A i)\"\n  proof\n    fix x\n    assume \"y = f x\"\n    assume xIA : \"x \u2208 (\u22c2 i \u2208 I. A i)\"\n    have \"f x \u2208 (\u22c2 i \u2208 I. f ` A i)\"\n    proof\n      fix i\n      assume \"i \u2208 I\"\n      with xIA have \"x \u2208 A i\" by simp\n      then show \"f x \u2208 f ` A i\" by simp\n    qed\n    with \u2039y = f x\u203a show \"y \u2208 (\u22c2 i \u2208 I. f ` A i)\" by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma \"f ` (\u22c2 i \u2208 I. A i) \u2286 (\u22c2 i \u2208 I. f ` A i)\"\n  by blast\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Esta semana he publicado en Calculemus las demostraciones con Lean4 e Isabelle\/HOL de las siguientes propiedades: 1. Uni\u00f3n con la imagen 2. Intersecci\u00f3n con la imagen 3. Uni\u00f3n con la imagen inversa 4. Imagen de la uni\u00f3n general 5. Imagen de la intersecci\u00f3n general A continuaci\u00f3n se muestran las soluciones.<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"default","_kad_post_title":"default","_kad_post_layout":"default","_kad_post_sidebar_id":"","_kad_post_content_style":"default","_kad_post_vertical_padding":"default","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[335],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":false,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8178"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/comments?post=8178"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8178\/revisions"}],"predecessor-version":[{"id":8180,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/8178\/revisions\/8180"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=8178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=8178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=8178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}