If M = x : 3 x < 8 and N = x : 8 < x 12 , which of the following is true? i. 8 M N ii. 8 M N iii. M N =

Analyze the question: We have two sets of real numbers (or integers) defined by inequalities. We must check which of the three statements are true.

Understand the sets:

• M = {x : 3 ≤ x < 8}. This includes 3 but excludes 8.
• N = {x : 8 < x ≤ 12}. This excludes 8 but includes 12.

Check each statement:

i. 8 ∈ M ∩ N 8 is not in M (because x < 8) and not in N (because x > 8). So 8 is not in the intersection. False.

ii. 8 ∈ M ∪ N For 8 to be in the union, it must be in M or in N. It is in neither. False.

iii. M ∩ N = ∅ M contains numbers less than 8, and N contains numbers greater than 8. No number can be both less than 8 and greater than 8, so they share no elements. The intersection is empty. True.

Conclusion: Only statement iii is true.

Correct answer: A (iii only)

Tip: Pay close attention to the inequality signs. "<" and ">" mean the boundary value is not included (open end), while "≤" and "≥" mean it is included (closed end). Whenever a boundary number such as 8 appears in both sets, check each set separately to see whether it really belongs. Here 8 is "missing" from both sets, so it is missing from both the union and the intersection.