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What are the domain and range of f? What is the inverse of f? Corppute fof fof. 1.4. REFERENCES AND SUGGESTED READINGS 24. Let f: X→Y be a map with A₁, A2 C X and B₁, B₂CY. (a) Prove f(A₁ U A₂) = f(A1) Uf(A₂). (b) Prove f(A₁A₂) c f(A1) nf(A₂). Give an example in which equality fails. (c) Prove f-¹(B₁ U B₂) = f(B₁) Uf-(B₂), where f¹(B) = (x EX: f(x) = B). (d) Prove f-¹(B₁ B₂) = f'(B₁) nf-¹(B₂). (e) Prove f-¹(Y\B₁)=X\f-¹(B₁). 25. Determine whether or not the following relations are equivalence relations on set. If the relation is an equivalence relation describe the partition given by

What are the domain and range of f? What is the inverse of f? Corppute fof fof. 1.4. REFERENCES AND SUGGESTED READINGS 24. Let f: X→Y be a map with A₁, A2 C X and B₁, B₂CY. (a) Prove f(A₁ U A₂) = f(A1) Uf(A₂). (b) Prove f(A₁A₂) c f(A1) nf(A₂). Give an example in which equality fails. (c) Prove f-¹(B₁ U B₂) = f(B₁) Uf-(B₂), where f¹(B) = (x EX: f(x) = B). (d) Prove f-¹(B₁ B₂) = f'(B₁) nf-¹(B₂). (e) Prove f-¹(Y\B₁)=X\f-¹(B₁). 25. Determine whether or not the following relations are equivalence relations on set. If the relation is an equivalence relation describe the partition given by

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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X z-1' What are the domain and range of f? What is the inverse of f? Corppute fof f-¹ of. 1.4. REFERENCES AND SUGGESTED READINGS 24. Let f: X→Y be a map with A₁, A2 C X and B₁, B₂ C Y. (a) Prove f(A₁ U A₂) = f(A₁) U f(A₂). (b) Prove f(A₁A₂) c f(A1) n f(A2). Give an example in which equality fails. (c) Prove f-¹(B₁ U B₂) = f'(B₁) Uf-¹(B₂), where f-¹(B) = {EX: f(x) e B). (d) Prove f-¹(B₁n B₂) = f'(B₁) nf-¹(B₂). (e) Prove f-¹(Y\ B₁)=X\f¹(B₁). 25. Determine whether or not the following relations are equivalence relations on t set. If the relation is an equivalence relation describe the partition given by
Transcribed Image Text:X z-1' What are the domain and range of f? What is the inverse of f? Corppute fof f-¹ of. 1.4. REFERENCES AND SUGGESTED READINGS 24. Let f: X→Y be a map with A₁, A2 C X and B₁, B₂ C Y. (a) Prove f(A₁ U A₂) = f(A₁) U f(A₂). (b) Prove f(A₁A₂) c f(A1) n f(A2). Give an example in which equality fails. (c) Prove f-¹(B₁ U B₂) = f'(B₁) Uf-¹(B₂), where f-¹(B) = {EX: f(x) e B). (d) Prove f-¹(B₁n B₂) = f'(B₁) nf-¹(B₂). (e) Prove f-¹(Y\ B₁)=X\f¹(B₁). 25. Determine whether or not the following relations are equivalence relations on t set. If the relation is an equivalence relation describe the partition given by
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