# Relative Difference between Infinite Set and Finite Set is Infinite

## Theorem

Let $S$ be an infinite set.

Let $T$ be a finite set.

Then $S \setminus T$ is an infinite set.

## Proof

Aiming for a contradiction, suppose $S \setminus T$ is a finite set.

Then:

 $\ds S$ $\subseteq$ $\ds S \cup T$ Set is Subset of Union $\ds$ $=$ $\ds \paren {S \setminus T} \cup T$ Set Difference Union Second Set is Union

By Union of Finite Sets is Finite, $S \cup T$ is a finite set.

By Subset of Finite Set is Finite, $S$ must also be a finite set.

But $S$ is an infinite set.

Therefore $S \setminus T$ is an infinite set.
$\blacksquare$