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Symmetric Relations

Symmetric relation in discrete mathematic between two or more elements of a set is such that if the first element is related to the second element, then the second element is also related to the first element as defined by the relation. As the name 'symmetric relations' suggests, the relation between any two elements of the set is symmetric. A symmetric relation is a binary relation.

There are different types of relations that we study in discrete mathematics such as reflexive, transitive, asymmetric, etc. In this lesson, we will understand the concept of symmetric relations and the formula to determine the number of symmetric relations along with some solved examples for a better understanding.

1.	What are Symmetric Relations?
2.	Asymmetric, Anti-symmetric and Symmetric Relations
3.	Number of Symmetric Relations
4.	FAQs on Symmetric Relations

What are Symmetric Relations?

Symmetric relation is defined In set theory as a binary relation R on X if and only if an element a is related to b, then b is also related to a for every a, b in X. Let us consider a mathematical example to understand the meaning of symmetric relations. Define a relation on the set of integers Z as 'a is related to b if and only if ab = ba'. We know that the multiplication of integers is commutative. So, if a is related to b, we have ab = ba ⇒ ba = ab, therefore b is also related to a and hence, the defined relation is symmetric.

Symmetric Relation Definition

Symmetric relation is a binary relation R defined on a set A for elements a, b ∈ A, we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R. This implies that a relation defined on a set A is a symmetric relation if and only if it satisfies aRb ⇔ bRa for all elements a, b in A. If there is a single ordered pair in R such that (a, b) ∈ R and (b, a) ∉ R, then R is not a symmetric relation.

Symmetric Relations

Examples of Symmetric Relations

'Is equal to' is a symmetric relation defined on a set A as if an element a = b, then b = a. aRb ⇒ a = b ⇒ b = a ⇒ bRa, for all a ∈ A
'Is comparable to' is a symmetric relation on a set of numbers as a is comparable to b if and only if b is comparable to a.
'Is a biological sibling' is a symmetric relation as if one person A is a biological sibling of another person B, then B is also a biological sibling of A.

Asymmetric, Anti-symmetric and Symmetric Relations

Asymmetric Relations - A relation R on a set A is said to be asymmetric if and only if (a, b) ∈ R, then (b, a) ∉ R, for all a, b ∈ A. In other words, asymmetric relation is the opposite of a symmetric relation. For example, the relation R defined as 'aRb if a is greater than b' on the set of natural numbers is an asymmetric relation as 15 > 10 but 10 is not greater than 15. Hence, (15, 10) ∈ R but (10, 15) ∉ R.
Antisymmetric relation - A relation R on a set A is said to be antisymmetric, if aRb and bRa hold if and only if when a = b. In other words, (a, b) ∉ R and (b, a) ∉ R if a ≠ b.
Symmetric Relation - A binary relation R defined on a set A is said to be symmetric iff, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R.

Number of Symmetric Relations

We can determine the number of symmetric relations on a set A. A relation R defined on a set A with n elements has ordered pairs of the form of (a, b). Now, we know that element 'a' can be chosen in n ways and similarly, element 'b' can be chosen in n ways. This implies we have n² ordered pairs (a, b) in R. Also, if (a, b) is in R, then for a symmetric relation, (b, a) is forced to be in R. Therefore, we have 2ⁿ^(n-1)/2 such ordered pairs. For a reflexive relation, we have ordered pairs of the form (a, a) which are also symmetric. We have 2ⁿ such ordered pairs. Hence, the number of symmetric relations is 2ⁿ. 2ⁿ^(n-1)/2 = 2ⁿ^(n+1)/2

Symmetric Relation Formula

Symmetric relations for a set having 'n' number of elements is given as N = 2ⁿ^(n+1)/2, where N is the number of symmetric relations and n is the number of elements in the set.

Related Topics to Symmetric relations

Important Notes on Symmetric Relations

A binary relation R defined on a set A is said to be symmetric iff, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R.
The number of symmetric relations on a set with the ‘n’ number of elements is given by 2^n(n+1)/2
A relation R on a set A is said to be asymmetric if and only if (a, b) ∈ R, then (b, a) ∉ R, for all a, b ∈ A.

Symmetric Relations Examples

Example 1: Suppose R is a relation on a set A where A = {1, 2, 3} and R = {(1,1), (1,2), (1,3), (2,3), (3,1)}. Check if R is a symmetric relation.

Solution: As we can see (1, 2) ∈ R. For R to be symmetric (2, 1) should be in R but (2, 1) ∉ R.

Hence, R is not a symmetric relation.

Answer: R = {(1,1), (1,2), (1,3), (2,3), (3,1)} is not a symmetric relation.
Example 2: Suppose R is a relation on a set A where A = {a, b, c} and R = {(a, a), (a, b), (a, c), (b, c), (c, a)}. Determine the elements which should be in R to make R a symmetric relation.

Solution: To make R a symmetric relation, we will check for each element in R.

(a, a) ∈ R ⇒ (a, a) ∈ R

(a, b) ∈ R ⇒ (b, a) ∈ R, but (b, a) ∉ R

(a, c) ∈ R ⇒ (c, a) ∈ R

(b, c) ∈ R ⇒ (c, b) ∈ R, but (c, b) ∉ R

Hence, (b, a) and (c, b) should belong to R to make R a symmetric relation.

Answer: (b, a) and (c, b) should belong to R to make R a symmetric relation.

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Practices Question on Symmetric Questions

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FAQs on Symmetric Relations

What are Symmetric Relations in Maths?

A binary relation R defined on a set A is said to be symmetric relation if and only if, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R.

What is the Formula for the Number of Symmetric Relations?

The number of symmetric relations on a set with the ‘n’ number of elements is given by 2^n(n+1)/2

Is Null Set a Symmetric Relation?

The null or empty set is a symmetric relation for every set. Since there are no elements in an empty set, the conditions for symmetric relation hold true.

What Are The Other Relations Similar To Symmetric Relation?

The other type of relations similar to symmetric relation is the reflexive relation and transitive relation. Further, the relation which is a symmetric relation, reflexive relation, and transitive relation is called an equivalence relation.

Is an Antisymmetric Relation always Symmetric Relation?

It is possible for a set to be symmetric and antisymmetric but not always. For example, R = {(1,1), (2, 2), (3, 3)} defined on A = {1, 2, 3} is symmetric as well as antisymmetric.

How to Tell if a Relation is a Symmetric Relation?

A binary relation R defined on a set A is said to be symmetric iff, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R.

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