Cardinality by inclusion–exclusion (divisible by 7 or 11 up to 30): Find |{x : x is a natural number ≤ 30 and divisible by 7 or 11}|.

Introduction / Context:Counting numbers with multiple divisibility conditions often uses inclusion–exclusion: count each condition separately, then subtract overlaps. Here, the ranges are small, so direct listing is fast and reliable.

Given Data / Assumptions:

• Natural numbers ≤ 30
• Divisible by 7 or 11

Concept / Approach:Count multiples of 7 up to 30 and multiples of 11 up to 30, then subtract the overlap (common multiples of 77 within the range) if any.

Step-by-Step Solution:Multiples of 7 ≤ 30: 7, 14, 21, 28 → 4 numbersMultiples of 11 ≤ 30: 11, 22 → 2 numbersOverlap lcm(7,11)=77 ≤ 30? No → 0 overlapTotal = 4 + 2 − 0 = 6

Verification / Alternative check:Sort the six values numerically to check: {7, 11, 14, 21, 22, 28}. No duplicates and all within 30.

Why Other Options Are Wrong:4 and 2 undercount by ignoring the other divisor; 8 double counts nonexistent overlaps; 2 is far too small.

Common Pitfalls:Assuming an overlap without checking the lcm bound; here 77 is outside the range.

Final Answer:6