Prove that Let f : A rightarrow B. Prove that if x A, Y B and f is a bijection.
ID: 2984769 • Letter: P
Question
Prove that
Let f : A rightarrow B. Prove that if x A, Y B and f is a bijection. then f(X) = Y iff f-1(Y) = X.Explanation / Answer
f is bijective => f is injective and surjective f:A?B IS a bijection. As f is a bijection, it is by definition an injection. So, by Inverse of Injection is Functional Relation?, f?1 is functional. Also, as f is a bijection, it is by definition a surjection. From Surjection iff Image equals Codomain, Image(f)=B. Thus from the definition of inverse relation, the domain of f?1 is: Dom(f?1)=B Thus we have established that: ?y?B : ?x?A : f?1(y)=x as Dom(f?1)=B ; ?y1 , y2 ? T : f?1(y1)?f?1(y2) => y1=y2 as f?1 is functional. Hence by definition f?1 is a mapping. Now suppose f?1 is a mapping. Then by definition: ?y1,y2?B : f?1(y1)?f?1(y2) =>y1=y2 which implies that: ?y1,y2?B : f(x1)=f(x2)=>x1=x2 and so f is an injection. From Preimage of Mapping equals Domain we have that: Im?1(f?1)=Dom(f?1) From the definition of inverse relation, the domain of f?1 is: Dom(f?1)=B From Surjection iff Image equals Codomain: Im(f)=B So f is a surjection. Being both an injection and a surjection (f-1) is a bijective which maps from b to a (f-1):B->A as x c A and y c B , (f-1)(y)=x
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