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Let k e Zt. Let A be a set such that |A| = k + 1, and let a e A (i.e., a is a fixed but unspecified element of A). Define X:= {X | X C A and a ¢ X} Y := {Y | Y C A and a e Y} 1. Show that XUY = P(A). Use as a theorem (i.e., assume that this is true, do not prove): If A, B,C are sets such that ACC and BCC, then AUBCC. 2. Show that Xny= Ø. 3. Show that X = P(A\ {a}) using a one-part proof.

Let k e Zt. Let A be a set such that |A| = k + 1, and let a e A (i.e., a is a fixed but unspecified element of A). Define X:= {X | X C A and a ¢ X} Y := {Y | Y C A and a e Y} 1. Show that XUY = P(A). Use as a theorem (i.e., assume that this is true, do not prove): If A, B,C are sets such that ACC and BCC, then AUBCC. 2. Show that Xny= Ø. 3. Show that X = P(A\ {a}) using a one-part proof.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 21E: 21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in...
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Let k e Z+. Let A be a set such that |A| = k + 1, and let a E A (i.e., a is a fixed but unspecified element of A). Define X := {X | X C A and a ¢ X} := {Y | Y C A and a e Y} Y 1. Show that XUY = P(A). Use as a theorem (i.e., assume that this is true, do not prove): If A, B,C are sets such that A CC and BC C, then AU BC C. 2. Show that XnY = Ø. 3. Show that X = P(A\ {a}) using a one-part proof. 4. Show that the function f : X → Y defined by f(X) = XU{a} VX€X is a bijection.
Transcribed Image Text:Let k e Z+. Let A be a set such that |A| = k + 1, and let a E A (i.e., a is a fixed but unspecified element of A). Define X := {X | X C A and a ¢ X} := {Y | Y C A and a e Y} Y 1. Show that XUY = P(A). Use as a theorem (i.e., assume that this is true, do not prove): If A, B,C are sets such that A CC and BC C, then AU BC C. 2. Show that XnY = Ø. 3. Show that X = P(A\ {a}) using a one-part proof. 4. Show that the function f : X → Y defined by f(X) = XU{a} VX€X is a bijection.
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