Question

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

Answer 1

Correct Answer: 960

Symmetric relation definition: A relation \( R \) is symmetric if for every pair \( (a, b) \) in \( R \), the pair \( (b, a) \) is also in \( R \). A relation is reflexive if \( (a, a) \) is in \( R \) for all \( a \).

Total number of relations:

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

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

Counting reflexive relations: Reflexive pairs require \( (1, 1), (2, 2), (3, 3), (4, 4) \) (4 pairs). The remaining potential symmetric pairs are \( (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4) \) (6 pairs).

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

Counting symmetric relations:

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

Non-reflexive symmetric relations:

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

Result: 960