Question

Prove the following statements: S Subset S Union T T Subset S Union T S Intersection T Subset S S Intersection T Subset T.

Answer 1

proofs:

S subset S Union T

We want to prove $S \subset S \cup T$

Let $s\in S$. By definition $S\cup T$ is the set that contains the elements of $T$ and the elements of $S$. Then $s$ must be in  $S\cup T$. As $s$ was arbitrary, we conclude that $S \subset S \cup T$.

T Subset S Union T

This proof is analogous to the previous one. In fact, this result is the same result as the previous one.

S Intersection T subset S

We want to prove $S \cap T \subset S$

Let $y\in S\cap T$. By definition of the intersection $y$ should be in $S$ and also in $T$. Then, we already saw that $y\in S$. As $y$ was arbitrary we can conclude that $S \cap T \subset S$.

S Intersection T subset T

This is the same result as the previous one. There is no need to prove it anymore, but if you wish, you can reply the exact same proof.