Given that P = {x : 1 $\leq x \leq 6$}, and Q = {x : 2 $\leq x \leq 10$} where x is an integer. Find n(P $\cap$ Q)

Analyze the question: We need the number of elements in the intersection of P and Q, where x is an integer.

Concept: P ∩ Q contains only the elements that belong to both sets. n(P ∩ Q) is the count of those elements.

Solution:

• P = {1, 2, 3, 4, 5, 6}
• Q = {2, 3, 4, 5, 6, 7, 8, 9, 10}

Elements common to both:

P ∩ Q = {2, 3, 4, 5, 6}

Counting them: n(P ∩ Q) = 5

You can also check this with the range method. The intersection runs from the larger lower bound (2) to the smaller upper bound (6), so it is 2 ≤ x ≤ 6. The number of integers is 6 − 2 + 1 = 5.

The correct option is A. 5.

Tip: For integer intervals, the intersection runs from the larger lower limit to the smaller upper limit. The count is (upper − lower + 1). Do not forget the "+1", because both endpoints are included.