Question

if $n(b) = 13$, $n(a cup b) = 22$, and $n(a cap b) = 6$, find $n(a)$.

Explanation:

Step1: Recall union formula

The principle of inclusion-exclusion for sets is:
$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

Step2: Rearrange to solve for $n(A)$

Isolate $n(A)$ by rearranging the formula:
$$n(A) = n(A \cup B) - n(B) + n(A \cap B)$$

Step3: Substitute given values

Plug in $n(A \cup B)=22$, $n(B)=13$, $n(A \cap B)=6$:
$$n(A) = 22 - 13 + 6$$

Step4: Calculate the result

Compute the arithmetic expression:
$$n(A) = 9 + 6 = 15$$

Answer:

15