Q.For $U = \{1, 2, 3, \dots, 9\}$, $A = \{2, 4, 6, 8\}$ and $B = \{1, 2, 3, 4\}$, find

Apply each definition to the listed elements; take the complement in (iv) relative to $U$.

$A \cap B = \{2,4\}$, $A \cup B = \{1,2,3,4,6,8\}$, $B - A = \{1,3\}$, $(A\cap B)' = \{1,3,5,6,7,8,9\}$.

(i) $A \cap B$ — common elements: $2$ and $4$ appear in both, so $A \cap B = \{2, 4\}$.

(ii) $A \cup B$ — all elements of either, listed once: $\{1, 2, 3, 4, 6, 8\}$.

(iii) $B - A$ — in $B$ but not $A$: remove $2, 4$ (which are in $A$) from $B = \{1,2,3,4\}$, leaving $\{1, 3\}$.

(iv) $(A \cap B)'$ — complement of $\{2, 4\}$ in $U = \{1, \dots, 9\}$: all of $U$ except $2$ and $4$, i.e. $\{1, 3, 5, 6, 7, 8, 9\}$.

For $B - A$, students sometimes compute $A - B$ instead (the two differences are not equal). For the complement, remember to exclude only the elements of $A \cap B$, taken against $U$.