*1.7 For any set X prove that ✗| is strictly smaller than |P(X)]. [This is evident when X is finite. When X is infinite the proof is surprisingly short but tricky.] 1.8 Let X, Y be non-empty sets. Prove that |X| > |Y| if and only if there exists a surjective function f: X → Y. 1.9 (a) Prove that a subset of a finite set is finite and that a subset of a countable set is countable. (b) Prove that the product of two and hence any finite number of countable sets is countable. [Hint: Note that the function f: Nx NN defined by f(x, y) = \(x + y − 1)(x + y − 2) + y for x, y E N is a bijection.]
*1.7 For any set X prove that ✗| is strictly smaller than |P(X)]. [This is evident when X is finite. When X is infinite the proof is surprisingly short but tricky.] 1.8 Let X, Y be non-empty sets. Prove that |X| > |Y| if and only if there exists a surjective function f: X → Y. 1.9 (a) Prove that a subset of a finite set is finite and that a subset of a countable set is countable. (b) Prove that the product of two and hence any finite number of countable sets is countable. [Hint: Note that the function f: Nx NN defined by f(x, y) = \(x + y − 1)(x + y − 2) + y for x, y E N is a bijection.]
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...
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![*1.7 For any set X prove that ✗| is strictly smaller than |P(X)]. [This is
evident when X is finite. When X is infinite the proof is surprisingly
short but tricky.]
1.8 Let X, Y be non-empty sets. Prove that |X| > |Y| if and only if there
exists a surjective function f: X → Y.
1.9 (a) Prove that a subset of a finite set is finite and that a subset of a
countable set is countable.
(b) Prove that the product of two and hence any finite number of
countable sets is countable. [Hint: Note that the function f: Nx
NN defined by f(x, y) = \(x + y − 1)(x + y − 2) + y for
x, y E N is a bijection.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faeff1758-93a2-4230-9942-96bb06e3cadc%2F47c271a6-8d31-4571-9193-717ec2787b15%2Fcb1334x_processed.jpeg&w=3840&q=75)
Transcribed Image Text:*1.7 For any set X prove that ✗| is strictly smaller than |P(X)]. [This is
evident when X is finite. When X is infinite the proof is surprisingly
short but tricky.]
1.8 Let X, Y be non-empty sets. Prove that |X| > |Y| if and only if there
exists a surjective function f: X → Y.
1.9 (a) Prove that a subset of a finite set is finite and that a subset of a
countable set is countable.
(b) Prove that the product of two and hence any finite number of
countable sets is countable. [Hint: Note that the function f: Nx
NN defined by f(x, y) = \(x + y − 1)(x + y − 2) + y for
x, y E N is a bijection.]
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