Set Equivalence

used to compare two or more sets, to see if equivalent.

Proving sets are equal

The ways to prove two sets $A$ and $B$ are equal:

• a Membership table
• a Containment_Proof do this later, never
• set comprehension and set equivalence
• Venn Diagram

Identity law:

• $P\cap U=P$
• $P\cup \varnothing =P$

Domination law:

• $P\cup U=U$
• $P\cap \varnothing =\varnothing$

Idempotent law:

• $P\cup P=P$
• $P\cap P=P$

Communitive law:

• $P\cup Q=Q\cup P$
• $P\cap Q=Q\cap P$

Associative law:

• $(P\cup Q)\cup R=P\cup (Q\cup R)$
• $(P\cap Q)\cap R=P\cap (Q\cap R)$

Distributive law:

• $P\cap (Q\cup R)=(P\cap Q)\cup (P\cap R)$
• $P\cup (Q\cap R)=(P\cup Q)\cap (P\cup R)$

De Morgan law:

• $\overline{P\cup Q}=\overline{P}\cap \overline{Q}$
• $\overline{P\cap Q}=\overline{P}\cup \overline{Q}$