Proving with a given definition that if $|A|=|B|$ then $A,B$ are equivalent (with induction but without using the induction hypothesis)

Question

Let $A,B$ be finite sets, we'll say the sets are equivalent if $|A\setminus B|=|B\setminus A|$.

Prove with the above definition that if $|A|=|B|$ then $A,B$ are equivalent.

Suppose $|A|=|B|=n$.

If $A\cap B =\emptyset$ then $A\setminus B= A, B\setminus A= B$ and we know they're equivalent.

Move to induction:

Suppose $A\cap B \neq\emptyset$ then there's a least one shared element.

So $|A\setminus B|=n-1=|B\setminus A|$

Suppose that the statement hold for all $k\in \mathbb N: k<n$ shared elements and we'll prove that it holds for $n$ shared elements.

If there are $n$ shared elements then $|A\setminus B|=0=|B\setminus A|$ and we're done.

Notice I didn't use the induction hypothesis, so is there actually no need for induction here? If so how this could be proved any differently?

Is it wrong to use induction without using the induction hypothesis?

Answer 1

We have, $|A|=|A\backslash B|+|A \cap B|$,

and $|B|=|B \backslash A|+|A \cap B|$
if $|A|=|B|$, then $|A\backslash B|+|A \cap B|=|B \backslash A|+|A \cap B|$.

We deduce that $|A\backslash B|=|B \backslash A|$

So, $A$ and $B$ are equivalent.

Ps. The inductive hypothesis is necessary to initialize the proof.