If a relation on a 3-element set is both reflexive and antisymmetric, how many such relations are possible?
Answer and explanation
Correct answer: 2^3
A reflexive relation must contain all three diagonal pairs (1,1), (2,2), and (3,3). For each of the three unordered off-diagonal pairs, such as {1,2}, antisymmetry allows either no ordered pair or exactly one of (1,2) and (2,1), giving two choices. Thus there are 2^3 possible relations. Option B incorrectly treats both directions as independently selectable, which would violate antisymmetry; the other counts do not match the three independent choices.
Frequently asked questions
What is the correct answer to this question?
2^3
Why is this the correct answer?
A reflexive relation must contain all three diagonal pairs (1,1), (2,2), and (3,3). For each of the three unordered off-diagonal pairs, such as {1,2}, antisymmetry allows either no ordered pair or exactly one of (1,2) and (2,1), giving two choices. Thus there are 2^3 possible relations. Option B incorrectly treats both directions as independently selectable, which would violate antisymmetry; the other counts do not match the three independent choices.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Types of relations.
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