Suppose a function $f : A → B$ is given. Define a relation $\sim$ on $A$ as follows: $a_1 \sim a_2 \iff f(a_1 ) = f(a_2 )$. Prove that $\sim$ is an equivalence relation on $A$.
I know that in order to prove this I need to prove reflexive, symmetry and transitivity. I'm not sure how to show that.
This should get you started. The definition of reflexivity says that for all $a\in A$, $a\equiv a$. Now, for the given relation, $a\equiv a$ would be shown to hold if $f(a)=f(a)$. Now, a function is required to return the same output on given input (this is part of the definition of function). Thus, $f(a)=f(a)$ always holds. So this verified that the condition for $a~a$ does hold, and thus that the relation is reflexive.
Now you will need to establish that it is also symmetric and transitive. Look at the definition of these concepts and proceed to prove them. For instance, your proof for symmetry could look something like this: Suppose that $a_1~\equiv a_2$. What does that mean (look at the definition of the relation!) Can you now conclude that $a_2 \equiv a_1$ (hint: yes, and then you proved symmetry!!).
