Commutative Laws of Sets

We will now look at the commutative laws between two sets. These proofs are relatively straightforward.

Theorem 1 (Commutative Law for the Union of Two Sets): If $A$ and $B$ are sets then $A \cup B = B \cup A$.

• Proof: Suppose that $x \in A \cup B$. Then $x \in A$ or $x \in B$ or $x \in A \cap B$ to which we write that $x \in B \cup A$. Therefore $A \cup B = B \cup A$. $\blacksquare$

Theorem 2 (Commutative Law for the Intersection of Two Sets): If $A$ and $B$ are sets then $A \cap B = B \cap A$.

• Proof: Suppose that $x \in A \cap B$. If $x$ is both in $A$ and $B$, then we can also say that $x$ is in both $B$ and $A$ or rather $x \in B \cap A$. Therefore $A \cap B = B \cap A$. $\blacksquare$