Show that if n(A△B) is even then n(A) and n(B) are even

Question

The question asked whether the statement is true or false and justify.

I am trying to use the identity n(A)+n(B)−2n(A∩B)=n(A△B) but can't find to know where to start or go

Answer 1

Write (assuming all the sets are finite)

$$A=\left(A\setminus B\right)\cup \left(A\cap B\right)\;,\;\;B=\left(B\setminus A\right)\cup \left(B\cap A\right)$$

and since each right side above is a disjoint union, we get

$$nA+nB=n(A\setminus B)+n(B\setminus A)+2n(A\cap B)$$

End now the argument using the definition of symmetric difference $\;A\Delta B\;$ ...