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

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Explanation:

as seen in the screenshot <3333^v^

Answer 2

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.