# Symmetric Relations: Definition, Formula, Examples, Facts, FAQs

A relation R defined on the set A is said to be symmetric, if (x, y) is an element of R, then (y, x) is also an element of R. In other words, if x is related to y, then y is also related to x.

Example: $R = \left\{(1,5), (5,1), (4,7), (7,4)\right\}$ defined on the set $A = \left\{1, 4, 5, 7\right\}$ is a symmetric relation.

A relation from a set A to a set B is a subset of the cartesian product A×B, and consists of ordered pairs (a,b), where a ∈ A and b ∈ B. If (a,b) ∈ R, we say that a is related to b. We can write this as aRb.

## Symmetric Relation Definition

The relation R defined on set A is symmetric if

(p, q) ∈ R ⇒ (q, p) ∈ R for all p, q ∈ A

pRq ⇒ qRp for all p, q ∈ A.

## How to Check if a Relation Is Symmetric

To check if a given relation is symmetric or not, we need to check if each ordered pair in the given relation satisfies the given condition.

(a, b) R (b, a) R

Example 1: Let $A = \left\{0, 1, 2\right\}$ and R be a relation defined on set A such that

$R = \left\{(0, 0), (1, 1), (2, 2), (1, 2)\right\}$. Is R symmetric?

We can see that the ordered pair (2, 1) is not present in R.

Thus, R is not symmetric.

Example 2: $A = \left\{x, y, z\right\}$ and $R = \left\{(x, y), (y, z), (x, z), (x,x), (y, x), (z, y), (z, x)\right\}$

Observe that for each ordered pair (a, b), the corresponding ordered pair (b, a) is present in R. Thus, R is symmetric.

## Number of Symmetric Relations

We can find the total number of symmetric relations on a given set with n elements.

Let us consider a set A with n elements on which a relation R is defined.

The total number of symmetric relations on A is given by $2^{\frac{n(n + 1)}{2}}$.

## Symmetric Relation Formula

Number of symmetric relations on a set with n elements $= N = 2^{\frac{n(n + 1)}{2}}$

where n is the number of elements in the set.

## Symmetric Relation Examples

• A commonly known example of symmetric relation is the relationship of biological siblings. If person X is a biological sibling to person Y, then Y is also the biological sibling of X. This is called the “is a biological sibling” symmetric relation.
• The “is equal to” relation is a symmetric relation since m = n, implies that n = m.
• The relation “is parallel to” defined on the set of straight lines is a symmetric relation since if a line l is parallel to m, we know that the line m is also parallel to the line l.

## Asymmetric, Antisymmetric, and Symmetric Relations

Let’s understand the difference between these three types of relations.

Symmetric Relations

A relation is symmetric if for all a, b A, (a,b) R ⇒ (b,a) R

Asymmetric Relations

A relation R on a set A is said to be asymmetric if and only if for all a, b A,

(a, b) R implies that (b, a) ∉ R,

In simple words, we can say that an asymmetric relation is the opposite of a symmetric relation.

Example: The relations “is less than (<)”, “is greater than (>)” are asymmetric relations.

Antisymmetric Relations

The relation R defined on set A is an antisymmetric relation if aRb and bRa implies that a = b.

Simply stated, if (a, b) R and a b, then (b, a) R.

## Facts on Symmetric Relation

• A relation can either be symmetric or antisymmetric but not the both.
• Even if a single ordered pair fails to meet the condition for the symmetric relation, the relation is considered not symmetric.
• Number of relations on set A having n elements $= 2^{|A \times A|} = 2^{n^{2}}$

## Conclusion

In this article, we learned about symmetric relations, how to find a symmetric relation, and the formula to find the number of symmetric relations on a set having n elements. Let’s solve a few examples based on these concepts.

## Solved Examples of Symmetric Relations

1. If R is a relation on a set $A = \left\{1, 2, 3\right\}$, where R is defined as

$R = \left\{(1,1), (1,2), (1,3), (2,3), (3,1)\right\}$, then check if R is a symmetric relation or not.

Solution:

For the relation R to be symmetric, we must have (b, a) ∈ R for each (a, b)∈ R.

As we can see that (1, 2)∈ R.

For R to be symmetric, (2, 1) should be in R, but that’s not the case.

Therefore, R is not a symmetric relation.

2. Let $R = \left\{(a, a), (e, e), (i, i), (o, o), (u, u)\right\}$ be a relation defined on the set $A = \left\{a, e, i , o , u\right\}$. Examine if R is symmetric.

Solution:

For the relation R to be symmetric, it should satisfy the following condition:

aRb implies that bRa.

In other words, if (a, b) ∈ R, then (b, a) ∈ R for all a, b in A.

In $R = \left\{(a, a), (e, e), (i, i), (o, o), (u, u)\right\}$, there is no ordered pair of the form (a, b). All ordered pairs are of the form (a, a). So, all ordered pairs satisfy this condition.

Hence, R is symmetric.

3. Let $A = \left\{p, q, r\right\}$ and R be a relation defined on the set A as shown:

$R = \left\{(p, p), (q, q), (p, r), (r, p), (r, r)\right\}$

Check if R is symmetric or not.

Solution:

To check if the relation R is symmetric or not, we check the condition given for each ordered pair in R.

(p, q) ∈ R ⇒ (q, p) ∈ R for all p, q ∈ A

Now, let us compare the above condition for every ordered pair in R.

Here, it can be seen that (p, r) ∈ R and also (r, p) ∈ R.

Hence, R is symmetric.

4. Let $A = \left\{1, 2, 3\right\}$ and $B = \left\{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\right\}$.

Show that B is symmetric or not.

Solution:

$A = \left\{1, 2, 3\right\}$ and $B = \left\{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\right\}$.

Here,

(2, 3) ∈ B but (3, 2)∉ B.

(1, 2) ∈ B but (2, 1) ∉ B.

So, B is not symmetric.

## Practice Problems on Symmetric Relations

1

### If A = {1, 2, 3}, the number of symmetric relations in A is

328
324
8
64
CorrectIncorrect
$N = 2^{\frac{n(n+1)}{2}} = 2^{\frac{3\times4}{2}} = 2^{6} = 64$
We get the number of symmetric relations as 64.
2

### The relation R = {(1, 2), (2, 1)} on set A = {1, 2} is

symmetric
asymmetric
reflexive
None of the above
CorrectIncorrect
The relation R is symmetric since for (1, 2) ∈ R, we have (2, 1) ∈ R and vice-versa.
3

aRa
bRb
bRa
a = b
CorrectIncorrect