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Question 17. Consider the function F : P(Z) → P(Z) defined by F(X)= X. For example, if X is the set of even integers, then F(X) = X be the set of odd integers. Determine if F is surjective, injective, or neither. Provide proofs or give counter-examples.

Question 17. Consider the function F : P(Z) → P(Z) defined by F(X)= X. For example, if X is the set of even integers, then F(X) = X be the set of odd integers. Determine if F is surjective, injective, or neither. Provide proofs or give counter-examples.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Question 17. Consider the function F : P(Z) → P(Z) defined by F(X) = X. For example, if X is the set of even integers, then F(X) = X be the set of odd integers. Determine if F is surjective, injective, or neither. Provide proofs or give counter-examples. A function having finite sets of the same size for the domain and codomain of a function means special things for injectivity and surjectivity. That is, if a function f : X → Y is one-to-one, with both X and Y finite and the same size (i.e., same number of elements), then f is also onto. And the converse is true as well. But this is no longer true when we have an infinite set like N or Z involved.
Transcribed Image Text:Question 17. Consider the function F : P(Z) → P(Z) defined by F(X) = X. For example, if X is the set of even integers, then F(X) = X be the set of odd integers. Determine if F is surjective, injective, or neither. Provide proofs or give counter-examples. A function having finite sets of the same size for the domain and codomain of a function means special things for injectivity and surjectivity. That is, if a function f : X → Y is one-to-one, with both X and Y finite and the same size (i.e., same number of elements), then f is also onto. And the converse is true as well. But this is no longer true when we have an infinite set like N or Z involved.
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