For a four-element set A, how many relations are symmetric but not reflexive?
Answer and explanation
Correct answer: 960
A symmetric relation on four elements has four diagonal pairs, each independently optional, and six unordered off-diagonal pairs, each offering two choices: include both directions or include neither. Thus there are 2^4×2^6=2^10 symmetric relations. To be non-reflexive, at least one diagonal pair must be absent, so subtract the 2^6 relations containing all four diagonals: 2^10−2^6=960. Option A is correct.
Frequently asked questions
What is the correct answer to this question?
960
Why is this the correct answer?
A symmetric relation on four elements has four diagonal pairs, each independently optional, and six unordered off-diagonal pairs, each offering two choices: include both directions or include neither. Thus there are 2^4×2^6=2^10 symmetric relations. To be non-reflexive, at least one diagonal pair must be absent, so subtract the 2^6 relations containing all four diagonals: 2^10−2^6=960. Option A is correct.
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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