Question

Consider a set \( S = \{ a, b, c, d \} \). Then the number of reflexive as well as symmetric relations from \( S \to S \) are

• A. 1024
• B. 256
• C. 16
• D. 64

Hint:

For reflexive and symmetric relations, remember to consider both the fixed reflexive pairs and the independent symmetric pairs where each has two possibilities.

Answer 1

The Correct Option is D

Step 1: Understand the concept of reflexive and symmetric relations.
A reflexive relation is a relation where every element is related to itself, i.e., \( (a, a), (b, b), (c, c), (d, d) \) must be included in the relation. A symmetric relation means that if \( (a, b) \) is in the relation, then \( (b, a) \) must also be in the relation. Step 2: Calculate the number of such relations.
The total number of possible relations from a set \( A \) to itself is given by \( n(A) \times n(A) = N^2 \), where \( N \) is the number of elements in the set. For \( N = 4 \), the total number of relations is \( 4^2 = 16 \). The reflexive relations require that each element is related to itself, leaving \( N^2 - N \) choices for the other elements. Each of these remaining relations has two choices: it can either be included or excluded from the relation. Hence, the number of reflexive and symmetric relations is: \[ 2^{N^2 - N} = 2^{4^2 - 4} = 2^{16 - 4} = 64 \]