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

To determine the number of reflexive and symmetric relations from the set \( S = \{ a, b, c, d \} \) onto itself, we need to understand two key concepts: reflexive and symmetric relations.

• Reflexive Relation: A relation \( R \) on a set \( S \) is reflexive if every element is related to itself. In other words, for each element \( x \) in \( S \), the pair \( (x, x) \) must be in \( R \). For the set \( S = \{ a, b, c, d \} \), the pairs that must be included for reflexivity are \( (a, a), (b, b), (c, c), (d, d) \).
• Symmetric Relation: A relation \( R \) on a set is symmetric if whenever \( (x, y) \) is in \( R \), then \( (y, x) \) is also in \( R \).

Now, let's count the number of reflexive and symmetric relations for the set \( S \):

1. For reflexivity, the relation must include the 4 specific pairs: \( (a, a), (b, b), (c, c), (d, d) \). So these pairs are fixed in the relation.
2. For symmetry, consider any pair \( (x, y) \) where \( x \neq y \). If \( (x, y) \) is part of the relation, then \( (y, x) \) must also be part of the relation to maintain symmetry.

Now, consider the off-diagonal pairs (pairs where the elements are different). For the set \( S \), these are:

• \((a, b)\) and \((b, a)\)
• \((a, c)\) and \((c, a)\)
• \((a, d)\) and \((d, a)\)
• \((b, c)\) and \((c, b)\)
• \((b, d)\) and \((d, b)\)
• \((c, d)\) and \((d, c)\)

Each off-diagonal pair choice is independent and can either be included in the relation or not. There are 6 such pairs, and each pair can be independently included or excluded, leading to \( 2^6 = 64 \) possibilities for symmetric relations.

Hence, the total number of relations that are both reflexive and symmetric is \(64\).

Therefore, the correct answer is 64.