# Help with proving some facts

• Nov 5th 2009, 11:45 AM
kaka87
Help with proving some facts
Hi,
I have to prove the following facts, but I am not sure if I figured them out correctly:

f : X --> Y and F : P(X) --> P(Y )

1) If A1 is a subset of A2, then F(A1) is a subset of F(A2)

For this one, I have the following:
For all x that are elements of A1 and A2
(there is a y that is an element of F(A1) s.t. F(x)=y) and (y is an element of F(A2) s.t. F(X)=y)

Therefore,
the fact that x is an element of A1 implies that there is a y that is an element of F(A2) s.t. F(x) = y

Hence, F(A1) is a subset of F(A2)

(Here I feel that, I missed a step constructing the proof with quantifiers.)

Also,

2) For every A that is an element of P(X), A is a subset of the inverse function of F(A)

For this one, I have:
(For all x that are elements of A, there is a y that is an element of F(A) s.t. F(x)=y s.t. the inverse of F(y)=x) implies that (for all x belonging to A, x is an element of the inverse of F(F(A))) implies that (A belongs to the inverse of F(F(A))).

I am also not sure about my logic here, do you guys agree with me?
• Nov 5th 2009, 07:30 PM
redsoxfan325
Quote:

Originally Posted by kaka87
Hi,
I have to prove the following facts, but I am not sure if I figured them out correctly:

f : X --> Y and F : P(X) --> P(Y )

1) If A1 is a subset of A2, then F(A1) is a subset of F(A2)

For this one, I have the following:
For all x that are elements of A1 and A2
(there is a y that is an element of F(A1) s.t. F(x)=y) and (y is an element of F(A2) s.t. F(X)=y)

Therefore,
the fact that x is an element of A1 implies that there is a y that is an element of F(A2) s.t. F(x) = y

Hence, F(A1) is a subset of F(A2)

(Here I feel that, I missed a step constructing the proof with quantifiers.)

Also,

2) For every A that is an element of P(X), A is a subset of the inverse function of F(A)

For this one, I have:
(For all x that are elements of A, there is a y that is an element of F(A) s.t. F(x)=y s.t. the inverse of F(y)=x) implies that (for all x belonging to A, x is an element of the inverse of F(F(A))) implies that (A belongs to the inverse of F(F(A))).

I am also not sure about my logic here, do you guys agree with me?

1. I think you have it, or if not, you're on the right track.

Let $\displaystyle y\in F(A_1)$. Then $\displaystyle \exists x\in A_1$ such that $\displaystyle F(x)=y$. Because $\displaystyle A_1\subset A_2$, $\displaystyle x\in A_2$ also. Thus $\displaystyle y=F(x)\in F(A_2)$, so $\displaystyle F(A_1)\subset F(A_2)$.

2. I'm not sure I understand the second question. It can't be asking you to prove that $\displaystyle A\subset F^{-1}(F(A))$, because that's trivial, but I don't know what else it could be asking.