Question

The number of symmetric relations defined on the set {1, 2, 3, 4} which are not reflexive is _____.

Answer 1

Correct Answer: 960

Define symmetric relations: A relation \( R \) is symmetric if \( (a, b) \in R \implies (b, a) \in R \). A relation is reflexive if \( (a, a) \in R \) for all \( a \).

Count total relations:

\[ \text{Total relations} = 2^{n^2} \text{ for } n = 4. \]

\[ \text{Total relations} = 2^{4^2} = 2^{16} = 65536. \]

Count reflexive relations: Reflexive pairs: \( (1, 1), (2, 2), (3, 3), (4, 4) \) (4 pairs). Remaining symmetric pairs: \( (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4) \) (6 pairs).

\[ \text{Total reflexive relations} = 2^6 = 64. \]

Count symmetric relations:

\[ \text{Symmetric relations} = 2^{\binom{n}{2} + n} = 2^{6 + 4} = 2^{10} = 1024. \]

Non-reflexive symmetric relations:

\(\text{Non-reflexive symmetric relations} = \text{Total symmetric relations} - \text{Reflexive symmetric relations} = 1024 - 64 = 960.\)

Thus, the answer is: 960