Question

The number of relations, defined on the set \{a, b, c, d\}, which 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

Step 1: Understanding the Question:
We are given a set \(A = \{a, b, c, d\}\) with 4 elements. A relation on \(A\) is a subset of the Cartesian product \(A \times A\). We need to find the count of relations that satisfy two properties simultaneously: reflexivity and symmetry.
Step 2: Key Formula or Approach:
Let the set be \(A\) with \(|A| = n\). The total number of elements in \(A \times A\) is \(n^2\).
- Reflexive Property: A relation \(R\) is reflexive if for every \(x \in A\), the pair \((x, x) \in R\). This means all diagonal elements of \(A \times A\) must be in \(R\).
- Symmetric Property: A relation \(R\) is symmetric if whenever \((x, y) \in R\), then \((y, x) \in R\).
We can count the number of such relations by considering the choices we have for including or excluding pairs of elements in the relation.
Step 3: Detailed Explanation:
The set \(A = \{a, b, c, d\}\), so \(n=4\). The set of all ordered pairs is \(A \times A\), which has \(n^2 = 16\) elements.
We can visualize these 16 pairs as a \(4 \times 4\) matrix:
\[ \begin{pmatrix} (a,a) & (a,b) & (a,c) & (a,d) \\ (b,a) & (b,b) & (b,c) & (b,d) \\ (c,a) & (c,b) & (c,c) & (c,d) \\ (d,a) & (d,b) & (d,c) & (d,d) \end{pmatrix} \] Condition 1: Reflexivity
For the relation to be reflexive, it must contain all the diagonal elements: \(\{(a,a), (b,b), (c,c), (d,d)\}\).
So, for these 4 pairs, we have no choice. They must be included in the relation. (1 choice for each)
Condition 2: Symmetry
For the relation to be symmetric, for any pair \((x, y)\) with \(x \neq y\), if \((x, y)\) is in the relation, then \((y, x)\) must also be in it.
This means we have to make decisions on pairs of off-diagonal elements \(\{(x, y), (y, x)\}\) together.
Let's count the number of such pairs. There are \(n^2 - n = 16 - 4 = 12\) off-diagonal elements.
These form \(\frac{n^2-n}{2} = \frac{12}{2} = 6\) pairs. For example, \(\{(a,b), (b,a)\}\) is one such pair.
For each of these 6 pairs, we have two choices:
1. Include neither \((x,y)\) nor \((y,x)\) in the relation.
2. Include both \((x,y)\) and \((y,x)\) in the relation.
So, for each of the 6 pairs of off-diagonal elements, we have 2 independent choices.
Total Number of Relations:
The total number of relations is the product of the number of choices for each group of elements.
- Choices for diagonal elements: \(1 \times 1 \times 1 \times 1 = 1^4 = 1\).
- Choices for off-diagonal pairs: \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6\).
Total number of reflexive and symmetric relations = \(1 \times 2^6 = 64\).
The general formula for a set of size \(n\) is \(2^{n(n-1)/2}\).
Step 4: Final Answer:
The total number of relations that are both reflexive and symmetric on the given set is 64.