Proofs with functions and subsets.

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Jamers328

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Sep 20, 2007
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Here is the problem I am having trouble with. It is for my Intro to Advanced Math class (Analysis, Proofs, etc).

f: A -> B and let C be a subset of A.

a.) Prove (or give a counterexample):
f(A\C) is a subset of f(A)\f(C)

b.) Prove (or give a counterexample):
f(A)/f(C) is a subset of f(A\C)

c.) What condition on f will ensure that f(A\C)=f(A)\f(C)? Prove this.

d.) What condition on f will ensure that f(A\C)=B\f(C)? Prove this.

I know we are supposed to show what work we have done, but I don't know what to do. I did draw a picture of A -> B and I drew C in A to try to understand what is going on with A, B, C, but I am lost on the proofs, and I'm not sure I even understand what c and d are asking.

Thanks in advance.
 
Have you written the a psrt as it should be? Is it f(A\B)?

For part b.
\(\displaystyle \begin{gathered} t \in f(A)\backslash f(C) \Rightarrow \quad t \in f(A) \wedge t \notin f(C) \hfill \\ \left( {\exists p \in A} \right)\left[ {f(p) = t \in f(A) \wedge f(p) \notin f(C)} \right] \hfill \\ \left( {p \in A \wedge p \notin C} \right) \Rightarrow \quad p \in A\backslash C \Rightarrow \quad t \in f(A\backslash C) \hfill \\ \therefore f(A)\backslash f(C) \subseteq f(A\backslash C). \hfill \\ \end{gathered}\)
 
No, I apologize. Part a is f(A\C) is a subset of f(A)\f(C).

Also, if the statement is false (which I believe a is false), we just have to give a counterexample. That is for both a and b.

I thought of a counterexample for part a. Can someone tell me if it is correct?
f(n)=|n-2|+1
A= 1,2,3
B= 4,5
C= 1,2

Thank you for b!
 
B must at least contain the range of f (i,e, f(A) C B). In that case, it wouldn't give a counterexample.

Try: f(n) = |n|
where A={-2,-1,0,1,2}, B=f(A), C={-2,-1,0}.
 
Oh ok... I didn't realize that.

Using my example then, could I just change B to {1,2}?

Using your example, B={0,1,2}, correct?
 
I like to keep it simple.
\(\displaystyle \begin{gathered} f = \left\{ {(1,a),(2,b),(3,c),(4,c),(5,b)} \right\} \hfill \\ A = \left\{ {1,2,3} \right\}\,\& \,C = \left\{ {3,4,5} \right\} \hfill \\ f\left( {A\backslash C} \right) = \left\{ {a,b} \right\}\,\& \,f(A)\backslash f(C) = \left\{ a \right\} \hfill \\ \end{gathered}\)
 
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