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2. The figure below represents the set X = (R – {0}) U {A, B} where A and B are two points not on the real line. A (a) Define f : X → R by Jo if x e {A, B} |x otherwise. f(x) = i. Is f injective? Prove or provide a counterexample. ii. Is f surjective? Prove or provide a counterexample. (b) For any two positive reals c < d, define IA(-c, d) = (-c, 0)U{A}U (0, d) IB(-c, d) = (-c, 0) U{B}U (0, d). i. Determine Y = U IA(-1/c, 1/d) and compute f(Y). c,dER>o ii. Determine Z = n IB(-1/c, 1/d) and compute f(Z). c,dER>o

2. The figure below represents the set X = (R – {0}) U {A, B} where A and B are two points not on the real line. A (a) Define f : X → R by Jo if x e {A, B} |x otherwise. f(x) = i. Is f injective? Prove or provide a counterexample. ii. Is f surjective? Prove or provide a counterexample. (b) For any two positive reals c < d, define IA(-c, d) = (-c, 0)U{A}U (0, d) IB(-c, d) = (-c, 0) U{B}U (0, d). i. Determine Y = U IA(-1/c, 1/d) and compute f(Y). c,dER>o ii. Determine Z = n IB(-1/c, 1/d) and compute f(Z). c,dER>o

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.2: Mappings
Problem 11E: For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions...
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2. The figure below represents the set X = (R – {0}) U {A, B} where A and B are two points not on the real line. A (a) Define f : X → R by Jo if x € {A, B} f (x) = otherwise. i. Is f injective? Prove or provide a counterexample. ii. Is f surjective? Prove or provide a counterexample. (b) For any two positive reals c < d, define IA(-c, d) = (-c, 0) U{A}U (0, d) IB(-c, d) = (-c, 0) U{B}U (0, d). i. Determine Y U IA(-1/c, 1/d) and compute f(Y). c,deR>o ii. Determine Z = IB(-1/c, 1/d) and compute f(Z). c,dER>o
Transcribed Image Text:2. The figure below represents the set X = (R – {0}) U {A, B} where A and B are two points not on the real line. A (a) Define f : X → R by Jo if x € {A, B} f (x) = otherwise. i. Is f injective? Prove or provide a counterexample. ii. Is f surjective? Prove or provide a counterexample. (b) For any two positive reals c < d, define IA(-c, d) = (-c, 0) U{A}U (0, d) IB(-c, d) = (-c, 0) U{B}U (0, d). i. Determine Y U IA(-1/c, 1/d) and compute f(Y). c,deR>o ii. Determine Z = IB(-1/c, 1/d) and compute f(Z). c,dER>o
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