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

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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?

(a, b)(b, a)Is (b, a) present in R?
(0, 0)(0, 0)Yes
(1, 1)(1, 1)Yes
(2, 2)(2, 2)Yes
(1, 2)(2, 1)No

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

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

Attend this quiz & Test your knowledge.

1

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

328
324
8
64
CorrectIncorrect
Correct answer is: 64
$N = 2^{\frac{n(n+1)}{2}} = 2^{\frac{3\times4}{2}} = 2^{6} = 64$
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
Correct answer is: symmetric
The relation R is symmetric since for (1, 2) ∈ R, we have (2, 1) ∈ R and vice-versa.
3

The relation R is said to be symmetric if aRb implies that

aRa
bRb
bRa
a = b
CorrectIncorrect
Correct answer is: aRa
The relation R is said to be symmetric if aRb implies that bRa.

Frequently Asked Questions on Symmetric Relations

In mathematics, this refers to the relationship between two or more elements such that if one element is related to another, then the other element is likewise related to the first element in a similar manner.

For the relation R defined from from A to B, the range is defined as the set of all second elements of the ordered pairs of R.

A relation is said to be an equivalence relation if it is reflexive, symmetric, and transitive.

The domain of the relation R is the set of all the first elements of the ordered pairs of R.

To prove that a relation is symmetric, you need to show that if an element a is related to an element b, then b is also related to a. For all pairs (a, b) that belongs to R, show that if (a, b) is in R, then (b, a) must also be in R.