Relative Difference between Infinite Set and Finite Set is Infinite

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Let $S$ be an infinite set.

Let $T$ be a finite set.

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


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


\(\ds S\) \(\subseteq\) \(\ds S \cup T\) 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.

This is a contradiction.

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