Binary Operation Intersection Proof

Question

Let * be a binary operation on set A. Let B and C be subsets of A that are closed under *.
A) prove that B ∩ C is closed under * B) Give example to show why B ∪ C is not always closed under *.

Answer 1

Suppose $a,b\in B\cap C$

$a*b\in B\\ a*b\in C$

therefore $a*b\in B\cap C$

Why is $B\cup C$ not closed? suppose $B = \mathbb Q[\sqrt 2], C = \mathbb Q[\sqrt 3]$

That is $B$ is the set numbers that can be expressed as $(p + q \sqrt2)$ where $p,q$ are rational

(And $C$ is the set numbers that can be expressed as $p + q \sqrt3$)

and $*$ is addition.

$B, C$ are closed under addition

but $\sqrt 2 + \sqrt 3 \notin B\cup C$