Science:Math Exam Resources/Courses/MATH220/December 2011/Question 04/Solution 1

We want to show that

$f(C\cap D)\subseteq f(C)\cap f(D)$

and that

$f(C)\cap f(D)\subseteq f(C\cap D)$

Let us start with the first.

Suppose that $y\in f(C\cap D)$; that is, there is some $x\in C\cap D$ such that $f(x)=y$. Since $x\in C$ and $x\in D$, it follows that $y\in f(C)$ and that $y\in f(D)$, i.e. that $y\in f(C)\cap f(D)$ as desired.

Note: This proof in no way uses injectivity; this is a general statement about all functions.

Now we prove the second statement. Suppose that $y\in f(C)\cap f(D)$. That is, $y\in f(C)$ and $y\in f(D)$. Equivalently, there is some $x_{1}\in C$ such that $f(x_{1})=y$, and some $x_{2}\in D$ such that $f(x_{2})=y$. We claim that $x_{1}=x_{2}$, and hence $x_{1}=x_{2}\in C\cap D$. But this follows immediately from the injectivity of $f$. It follows now, since $f(x_{1})=y$ and $x_{1}\in C\cap D$ that $y\in f(C\cap D)$ as claimed.