Answer
a. 30
b. 29
c. 24
d. 18
Work Step by Step
$|A\cup B|=|A|+|B|-|A\cap B|$, $|A|=12$, $|B|=18$
a. If $A\cap B=\emptyset$, then $|A\cup B|=0$. Using the Principle of Inclusion-Exclusion, $|A\cup B|=|A|+|B|-|A\cap B|=12+18-0=30$.
b. $|A\cap B|=1$: Using the Principle of Inclusion-Exclusion, $|A\cup B|=|A|+|B|-|A\cap B|=12+18-1=29$.
c. $|A\cap B|=6$: Using the Principle of Inclusion-Exclusion, $|A\cup B|=|A|+|B|-|A\cap B|=12+18-6=24$.
d. $A \subseteq B$ means that every element of $A$ is also an element of $B$, so $|A\cap B|=|A|=12$. Using the Principle of Inclusion-Exclusion, $|A\cup B|=|A|+|B|-|A\cap B|=12+18-12=18$.
