If A has 3 elements, how many symmetric relations on A are also reflexive?
Answer and explanation
Correct answer: 8
Answer: B, 8. Reflexivity fixes all three diagonal pairs: (1,1), (2,2), and (3,3) must be present. For symmetry, an off-diagonal pair must occur together with its reverse. The three independent unordered pairs are {1,2}, {1,3}, and {2,3}. For each one, we have two choices: include both ordered versions or include neither. Thus the number of choices is 2×2×2=2^3=8. Option A misses one independent choice, while option C counts four independent choices that do not exist. Option D treats all six off-diagonal ordered pairs as independent, but symmetry links them in reverse-direction pairs. Memory cue: diagonal pairs are fixed by reflexivity; only one decision is made for each unordered pair of different elements.
Frequently asked questions
What is the correct answer to this question?
8
Why is this the correct answer?
Answer: B, 8. Reflexivity fixes all three diagonal pairs: (1,1), (2,2), and (3,3) must be present. For symmetry, an off-diagonal pair must occur together with its reverse. The three independent unordered pairs are {1,2}, {1,3}, and {2,3}. For each one, we have two choices: include both ordered versions or include neither. Thus the number of choices is 2×2×2=2^3=8. Option A misses one independent choice, while option C counts four independent choices that do not exist. Option D treats all six off-diagonal ordered pairs as independent, but symmetry links them in reverse-direction pairs. Memory cue: diagonal pairs are fixed by reflexivity; only one decision is made for each unordered pair of different elements.
Which subject and chapter does this question cover?
This is a Class 11 Mathematics question. Chapter: Relations and Functions. Topic: Relations that are reflexive and symmetric.
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