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If A and B are sets and f: A→ B, then for any subset S of A we define f(S) = {be B: b= f(a) for some a € S}. Similarly, for any subset T of B we define the pre-image of T as f(T) = {ae A: f(a) e T}. Note that f-¹(T) is well defined even if f does not have an inverse! For each of the following state whether it is True or False. If True then give a proof. If False then give a counterexample: (a) f(S₁US₂) = f(S₁) u f(S₂) (b) f(Sin S₂) = f(S₁) nf (S₂) (c) f¹(T₁UT₂) = f¹(T₁)uf-¹(T₂) (d) f-¹(T₁T₂) = f-¹(T₁) nf-¹(T₂)

If A and B are sets and f: A→ B, then for any subset S of A we define f(S) = {be B: b= f(a) for some a € S}. Similarly, for any subset T of B we define the pre-image of T as f(T) = {ae A: f(a) e T}. Note that f-¹(T) is well defined even if f does not have an inverse! For each of the following state whether it is True or False. If True then give a proof. If False then give a counterexample: (a) f(S₁US₂) = f(S₁) u f(S₂) (b) f(Sin S₂) = f(S₁) nf (S₂) (c) f¹(T₁UT₂) = f¹(T₁)uf-¹(T₂) (d) f-¹(T₁T₂) = f-¹(T₁) nf-¹(T₂)

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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If A and B are sets and f: A→ B, then for any subset S of A we define f(S) = {be B: b= f(a) for some a € S}. Similarly, for any subset T of B we define the pre-image of T as f(T) = {ae A: f(a) e T}. Note that f-¹(T) is well defined even if f does not have an inverse! For each of the following state whether it is True or False. If True then give a proof. If False then give a counterexample: (a) f(S₁US₂) = f(S₁) u f(S₂) (b) f(Sin S₂) = f(S₁) nf (S₂) (c) f¹(T₁UT₂) = f¹(T₁)uf-¹(T₂) (d) f-¹(T₁T₂) = f-¹(T₁) nf-¹(T₂)
Transcribed Image Text:If A and B are sets and f: A→ B, then for any subset S of A we define f(S) = {be B: b= f(a) for some a € S}. Similarly, for any subset T of B we define the pre-image of T as f(T) = {ae A: f(a) e T}. Note that f-¹(T) is well defined even if f does not have an inverse! For each of the following state whether it is True or False. If True then give a proof. If False then give a counterexample: (a) f(S₁US₂) = f(S₁) u f(S₂) (b) f(Sin S₂) = f(S₁) nf (S₂) (c) f¹(T₁UT₂) = f¹(T₁)uf-¹(T₂) (d) f-¹(T₁T₂) = f-¹(T₁) nf-¹(T₂)
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