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Abstract Algebra

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Moderate

Closure of Set Operations

ABSALG-JRXVKS

Let $X$ be any set and $S=P(X)$ be the power set of $X$, i.e. $S$ is the set of all subsets of $X$. Identify which of the following set operations are closed on $S$.

Select ALL that apply.

A

Intersection: $(A,B)\mapsto A\cap B=\{x\in X: x\in A \textrm{ and } x\in B\}$.

B

Union: $(A,B)\mapsto A\cup B=\{x\in X: x\in A \textrm{ or } x\in B\}$.

C

Relative Complement: $(A,B)\mapsto A- B=\{x\in X: x\in A \textrm{ and } x\not\in B\}$.

D

Cartesian Product: $(A,B)\mapsto A\times B=\{(a,b):a\in A \textrm{ and } b\in B\}$.

E

None of the above.