Question

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

• A. \(1024\)
• B. \(256\)
• C. \(16\)
• D. \(64\)

Hint:

For relations on a set of \( n \) elements: \begin{itemize} \item Reflexive relations require all \( (a,a) \) to be included \item Symmetric relations treat \( (a,b) \) and \( (b,a) \) as a single choice \item Number of symmetric choices \( = 2^{\frac{n(n-1)}{2}} \) \end{itemize}

Answer 1

The Correct Option is D

Given: A set \( S = \{a, b, c, d\} \).

Step 1: Reflexive Relation

A relation is reflexive if for every element \( x \in S \), the pair \( (x, x) \) is included. So, the relation must include the following reflexive pairs:

\( (a, a), (b, b), (c, c), (d, d) \).

These are fixed and must always be in the relation.

Step 2: Symmetric Relation

A relation is symmetric if for every pair \( (x, y) \), if \( (x, y) \) is in the relation, then \( (y, x) \) must also be included.

For the non-reflexive pairs, we need to ensure symmetry:

• Pairs like \( (a, b), (a, c), (a, d), (b, c), (b, d), (c, d) \) must be handled symmetrically.

For each of these 6 pairs, we have 2 choices:

• Either include both \( (x, y) \) and \( (y, x) \) in the relation.
• Or exclude both \( (x, y) \) and \( (y, x) \) from the relation.

Step 3: Total Number of Relations

Since there are 6 non-reflexive pairs, and for each pair, we have 2 choices, the total number of reflexive and symmetric relations is:

\( 2^6 = 64 \).

Final Answer: The number of reflexive and symmetric relations from \( S \to S \) is \( \boxed{64} \).