The number of relations, defined on the set {a, b, c, d}, which are both reflexive and symmetric, is

Q: Why do we only care about the non-diagonal elements for the symmetric part?

Great question! The diagonal elements $(x, x)$ are already fixed by the reflexivity condition. Since $(x, x)$ is its own mirror image, it doesn't impose any additional constraints on other pairs. The symmetry condition only restricts the non-diagonal elements, which must appear in pairs $(x, y)$ and $(y, x)$.

Q: What if the question asked for reflexive, symmetric, AND transitive relations? Would the answer be different?

Absolutely! That would change the problem entirely. A relation that is reflexive, symmetric, and transitive is an equivalence relation. Counting those is much more complex and involves Bell numbers. For a set of size $n=4$, the number of equivalence relations is the 4th Bell number, which is 15.

Q: Is there a way to visualize this without the grid?

Yes! Think of the set as nodes in a graph. Reflexivity means every node has a self-loop. Symmetry means if there is an edge from node $x$ to $y$, there must be an edge from $y$ to $x$. You are essentially deciding whether to draw an undirected edge between any two distinct nodes. There are $\frac{n(n-1)}{2}$ possible undirected edges, and for each, you have 2 choices (edge or no edge).

Question

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

Options:

(A) (1) 16

(B) (2) 1024

(C) (3) 64

(D) (4) 256

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are an architect designing a structure on a set $A = \{a, b, c, d\}$. You are building a mathematical relation, which is defined as a subset of the Cartesian product $A 	imes A$.

Since $A$ has $n=4$ elements, the Cartesian product $A 	imes A$ contains $n^2 = 16$ possible ordered pairs. Think of this as a $4 	imes 4$ grid where each cell $(x, y)$ represents a potential connection.

### The Reflexive Lock

The problem imposes a strict condition: the relation must be reflexive. This means for every element $x$ in $A$, the pair $(x, x)$ must be in $R$.

In our grid, these are the diagonal elements: $(a, a), (b, b), (c, c),$ and $(d, d)$. Because these are mandatory, we have no choice but to include them.

They are locked in. We have effectively filled 4 cells out of 16, leaving us with $16 - 4 = 12$ cells to decide upon.

### The Dance of Symmetry

Now, we introduce the second constraint: symmetry. A relation is symmetric if $(x, y) \in R$ implies $(y, x) \in R$.

This is a restrictive rule. It means that if you decide to include a connection from $a$ to $b$, you are forced by the laws of symmetry to also include the connection from $b$ to $a$.

They are a package deal. If you look at our remaining 12 non-diagonal cells, they naturally form 6 pairs of mirror images: $(a, b)$ with $(b, a)$, $(a, c)$ with $(c, a)$, and so on. We have 6 independent 'mirror pairs' to make decisions about.

### The Power of Choice

For each of these 6 independent pairs, you have exactly two choices: either you include both elements in your relation, or you exclude both. There is no middle ground.

Since we have 6 independent decisions to make, and each decision has 2 options, the total number of ways to form such a relation is:

$$2 	imes 2 	imes 2 	imes 2 	imes 2 	imes 2 = 2^6$$

Calculating this, we get **64**.

### The General Elegance

This logic is a universal truth. For any set with $n$ elements, the number of reflexive and symmetric relations is given by the formula:

$$2^{\frac{n(n-1)}{2}}$$

When you plug in $n=4$, you get:

$$2^{\frac{4 	imes 3}{2}} = 2^6 = 64$$

You have just mastered a core concept of discrete mathematics. The next time you see a relation problem, don't just calculate; visualize the grid, lock the diagonal, and watch the symmetric pairs dance.

Animated Solution for Mathematics - Sets and Relations: The number of relations, defined on the set ${a, b, c, d}$ , which are both reflexive and symmetric, is equal to:

Select Answer:

Visualized Solution

Visualizing the Set and Cartesian Product

The Reflexive Condition

Choices for Diagonal Elements

The Symmetric Condition

Visualizing Symmetric Pairs

Counting the Pairs

Choices for Each Pair

Final Calculation

General Formula

The Sigma Insight: Types of Relations

Analyzing the Setup

The Reflexive Lock

The Dance of Symmetry

The Power of Choice

The General Elegance

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Animated Solution for Mathematics - Sets and Relations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of relations, defined on the set {a,b,c,d}, which are both reflexive and symmetric, is equal to: