Cardinality of an even-number set with bounds: Let S = {x : x = 2n, n ∈ N, 4 ≤ x ≤ 11}. What is |S| (the number of elements)?

Introduction / Context:This is a count of even numbers in a closed interval. Express x = 2n and list feasible x within [4, 11].

Given Data / Assumptions:

• x = 2n with n ∈ N = {1,2,3,...}
• 4 ≤ x ≤ 11

Concept / Approach:Enumerate the even numbers between 4 and 11 inclusive, or use arithmetic progression logic to count terms quickly.

Step-by-Step Solution:Valid x: 4, 6, 8, 10Count = 4

Verification / Alternative check:AP method: start a=4, difference d=2, last l=10. Number of terms k = ((l−a)/d) + 1 = ((10−4)/2) + 1 = 3 + 1 = 4.

Why Other Options Are Wrong:6, 8, 12 overcount; 3 undercounts by missing one even number.

Common Pitfalls:Accidentally including 12 (outside the bound) or excluding one endpoint unnecessarily.

Final Answer:4