Question

The number of relations defined on the set \( \{a, b, c, d\} \) that are both reflexive and symmetric is equal to:

• A. 1024
• B. 64
• C. 16
• D. 256

Hint:

To count relations with specific properties, break down the set of all possible pairs (\(A \times A\)) into groups based on the properties.
For reflexive and symmetric relations on a set of size \(n\):
- Diagonal pairs \((x,x)\): \(n\) pairs, choice is fixed by reflexivity (must be included or excluded).
- Off-diagonal pairs \(\{(x,y), (y,x)\}\): \(n(n-1)/2\) such pairs. For symmetry, the choice for \((x,y)\) and \((y,x)\) are linked. You can either have both or none. So 2 choices for each pair.
The total is the product of these choices.

Answer 1

The Correct Option is B

To find the number of relations on the set \(\{a, b, c, d\}\) which are both reflexive and symmetric, we need to understand the properties of reflexive and symmetric relations. 

Reflexive Relation: A relation \( R \) on a set \( S \) is reflexive if every element is related to itself. For our set \(S = \{a, b, c, d\}\), the reflexive pairs are: \((a, a), (b, b), (c, c), (d, d)\). These pairs must be included in every reflexive relation.

Symmetric Relation: A relation \( R \) is symmetric if whenever \( (x, y) \in R \), then \( (y, x) \in R \) for all \( x, y \) in the set.

Steps to Calculate the Number of Relations:

1. \(4\) pairs are already fixed due to the reflexive property: \((a, a), (b, b), (c, c), (d, d)\).
2. Symmetric pairs involve combinations of remaining elements \((\{a, b\}, \{a, c\}, \{a, d\}, \{b, c\}, \{b, d\}, \{c, d\})\). Each choice affects two entries, e.g., \((a, b)\) and \((b, a)\).
3. There are \(\binom{4}{2} = 6\)possible unordered pairs.
4. Each of these symmetric pairs can be either included or not included in a relation, leading to \(2^6\) possible combinations.

Thus, the number of reflexive symmetric relations on the set \(S = \{a, b, c, d\}\) is:

\(2^6 = 64\).

Hence, the correct answer is 64.