Question

if $n(a) = 9$, $n(a cup b) = 11$, and $n(b) = 5$, then what is $n(a cap b)$?
resources
ebook
submit answer
/ 6 points
refer to the following venn diagram.
venn diagram with two overlapping circles a and b, circle a has 12, the overlap has 3, circle b has 15, and outside both circles has 20 (partially visible)

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

Compute the arithmetic expression:
$$n(A \cap B) = 3$$

Answer:

3