Let A and B be sets and let f : A → B be a function. Show that if D

gorgeousgen9487 2022-07-03 Answered
Let A and B be sets and let f : A B be a function. Show that if D B, then f ( f 1 ( D ) ) D.
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Answers (2)

Tamia Padilla
Answered 2022-07-04 Author has 16 answers
Step 1
Some things that may help. Say we have a function F : A B and that b B (b is an element of B). Then, f 1 ( b ) is a set (yes, a set, I know that can be confusing).
The elements of the set f 1 ( b ) are the elements a A such that f ( a ) = b.
If you know set builder notation, then we say that f 1 ( b ) = { a A : f ( a ) = b }.
Now, to generalize this, let's say instead of a point in B, we have a subset of B, call it D. Then, f 1 ( D ) = { a A : f ( a ) D }.
Step 2
And, similarly, if C A, then f(C) is also a set (this time it's f that's used to make a set, not f 1 ). Precisely, f ( C ) = { f ( c ) : c C }.
So, if x f ( f 1 ( D ) ), we replace C with f 1 ( D )
f ( f 1 ( D ) ) = { f ( c ) : c f 1 ( D ) }
This should help you get started. Let me know if you have more questions.
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mistergoneo7
Answered 2022-07-05 Author has 3 answers
Step 1
This should help: For every function f : A B (whether bijective, injective, surjective, or none of these), we can defined a function f 1 : P ( B ) P ( A ) Y { x A f ( x ) Y }
Step 2
But also f : P ( A ) P ( B ) X { f ( x ) x X } .
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