Question

if ( n(a) = 9 ), ( n(a cup b) = 11 ), and ( n(b) = 5 ), then what is ( n(a cap b) )?

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 $n(A \cap B)$

Isolate the unknown term:
$$n(A \cap B) = n(A) + n(B) - n(A \cup B)$$

Step3: Substitute given values

Plug in $n(A)=9$, $n(B)=5$, $n(A \cup B)=11$:
$$n(A \cap B) = 9 + 5 - 11$$

Step4: Calculate the result

Perform arithmetic operations:
$$n(A \cap B) = 14 - 11 = 3$$

Answer:

3