how to prove this: $f(A)=B$

Question

I am given two sets: $A$ and $B$ and a function $f: A \rightarrow B$.

I am asked to show and prove whether $f(A)=B$ is true or false.

I am stuck not knowing how to do this.

How can I do this?

Answer 1

The standard way of proving such a thing is to pick an arbitrary element $b \in B$ and find some $a \in A$ so that $f(a)=b$.

For example, suppose $A= \mathbb{Z}, B= \mathbb{Z}$ and $f(x)=x+5$. We pick some arbitrary element $z \in \mathbb{Z}$. Then we observe that $f(z-5)=(z-5)+5=z$ and $z-5 \in \mathbb{Z}$. This shows that $f(A)=B$.

In general such a statement may be true or false. For instance, if we take $A, B$ as in the above example, then the function $f(x)=2x$ does not satisfy this property since there is no $x$ such that $f(x)=3$.

Answer 2

Take $A = B = \{0,1\}$ and $f(x) = 0$ for $x=0,1$. This shows it is false in general.