Prove Theorem 5.5.6(2). Let X and Y be sets, A, B subseteq...

Prove Theorem 5.5.6(2). Let $X$ and $Y$ be sets, $A, B subseteq X$, and $f : X o Y$. (a) Prove that $f[A cap B] subseteq f[A] cap f[B]$. (b) Give an example of sets $X$ and $Y$, subsets $A, B subseteq X$, and a function $f : X o Y$ such that $f[A cap B] eq f[A] cap f[B]$.

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Transcript

00:01 Hello everyone we are going to solve a question in this question we are given that a and b are subset of x and there we define a mapping f from x to y and in a part we have to prove that function that is f of a intersection b is subset of here we have subset of f of a intersection f of b.

00:29 So this we have to prove.

00:32 So let's start with the solution of this.

00:35 Firstly, we let y, which is in f a intersection b.

00:42 So that means there exists, this implies there exist.

00:48 X that belongs to x uniting.

00:52 X union b here we have they define x which belongs to we can say a union b such that f of x equals to y.

01:10 So this means that x belongs to a and x belongs to b because x belongs to a intersection b.

01:22 So from here we can say that f of x belongs to f of a and f of x belongs to f of b.

01:32 So this means that f of x belongs to f of a intersection f of b.

01:41 And this f of x is y.

01:43 So from here we have y belongs to f of a intersection f of b.

01:48 But we take y in f a intersection b.

01:52 So from here we have that f of a intersection b is subset of f of a intersection f of b.

02:00 So this is the proof of a part.

02:03 Now moving on to the b part.

02:05 In this part we have to give the example of x and y...

Question

Prove Theorem 5.5.6(2). Let $X$ and $Y$ be sets, $A, B subseteq X$, and $f : X o Y$. (a) Prove that $f[A cap B] subseteq f[A] cap f[B]$. (b) Give an example of sets $X$ and $Y$, subsets $A, B subseteq X$, and a function $f : X o Y$ such that $f[A cap B] eq f[A] cap f[B]$.

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