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5. Given a function f : A → B. Define the relation f from P(A) to P(B) by f = {(X,Y) C P(A) × P(B) | Y = f(X)}. |3D a. Show that f is a function from P(A) to P(B). b. Show that if f is one-to-one, then f is one-to-one. c. Show that if f is onto B, then f is onto P(B).

5. Given a function f : A → B. Define the relation f from P(A) to P(B) by f = {(X,Y) C P(A) × P(B) | Y = f(X)}. |3D a. Show that f is a function from P(A) to P(B). b. Show that if f is one-to-one, then f is one-to-one. c. Show that if f is onto B, then f is onto P(B).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Write all answers clearly. Proofs should be CLEAR, COHERENT, COMPLETE, and CONCISE. Let A, B, and C be sets. 1. Prove: If AC B then A NCCBNC. 2. Disprove: If AnCC BnC, then A C B. 3. Disprove: For any relations R and S, ran(S o R) = ran(S). 4. Prove or disprove: If R and S are equivalence relations on A, then S o R is an equivalence relation on A. Justify your answer. 5. Given a function f : A → B. Define the relation f from P(A) to P(B) by f = {(X,Y) C P(A) × P(B) | Y = f(X)}. a. Show that f is a function from P(A) to P(B). b. Show that if f is one-to-one, then f is one-to-one. c. Show that if f is onto B, then f is onto P(B).
Transcribed Image Text:Write all answers clearly. Proofs should be CLEAR, COHERENT, COMPLETE, and CONCISE. Let A, B, and C be sets. 1. Prove: If AC B then A NCCBNC. 2. Disprove: If AnCC BnC, then A C B. 3. Disprove: For any relations R and S, ran(S o R) = ran(S). 4. Prove or disprove: If R and S are equivalence relations on A, then S o R is an equivalence relation on A. Justify your answer. 5. Given a function f : A → B. Define the relation f from P(A) to P(B) by f = {(X,Y) C P(A) × P(B) | Y = f(X)}. a. Show that f is a function from P(A) to P(B). b. Show that if f is one-to-one, then f is one-to-one. c. Show that if f is onto B, then f is onto P(B).
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