Types of relation- Symmetric

A symmetric relation between two or more elements of set-in discrete mathematics is one in which if the first element is connected to the second element, the second element is also related to the first element as described by the relation. The relationship between any two items of the set is symmetric, as the term ‘symmetric relations’ implies. A binary relation is a symmetric relation. In discrete mathematics, we investigate several sorts of relations such as reflexive, transitive, asymmetric, and so on. In this lesson, we will learn about symmetric definition and the formula for calculating the number of symmetric relations, as well as some solved instances to help us understand the concept.

What are Symmetric Relations?

In set theory, a binary relation R on X is said to be symmetric if and only if, for every a, b in X, an element an is related to b, and vice versa. Let’s look at a mathematical example to better grasp what symmetric relations are. Define the relation ‘a is connected to b if and only if ab = ba’ on the set of integers Z. Integer multiplication is commutative, as we all know. So, if an is connected to b, we obtain ab = ba ⇒ ba = ab, which means b is related to an as well, and thus the stated relation is symmetric.

Symmetric Relation Formula

N = 2n(n+1)/2, where N is the number of symmetric relations and n is the number of items in the set, gives the number of symmetric relations on a set with ‘n’ elements.

Examples of Symmetric Relations

After learning about what is symmetric, let’s take a look at its examples.

• The symmetric relation ‘is equal to’ is defined on a set A as follows: if an element a = b, then b = a. aRb ⇒ a = b ⇒ b = a ⇒ bRa, for all a ∈ A

• The symmetric relation ‘is comparable to’ on a set of numbers states that an is comparable to b if and only if b is comparable to a.

• If one individual A is a biological sibling of another person B, then B is likewise a biological sibling of A.

Know the Difference Between Asymmetric, Anti-symmetric and Symmetric Relations

Asymmetric Relations – 

If and only if (a, b) ∈ R, then (b, a) ∉ R, for all a, b ∈ A, a relation R on a set A is said to be asymmetric. To put it another way, an asymmetric relation is the polar opposite of a symmetric one. For example, on the set of natural numbers, the relation R defined as ‘aRb if an is bigger than b’ is an asymmetric relation because 15 > 10 but 10 is not greater than 15. As a result, (15, 10) ∈ R, but (10, 15) ∉ R.

Antisymmetric relation – 

If aRb and bRa hold if and only if a = b, a relation R on a set A is said to be antisymmetric. In other words, it could be termed as (a, b) ∉ R and (b, a) ∉ R if a ≠ b.

Symmetric Relation – 

On a set A, a binary relation R is said to be symmetric if we have aRb, that is, (a, b) R, for members a, b ∈ A, and then we must have aRb, that is, (a, b) ∈ R

Number of Symmetric Relations

On a set A, we can count the number of symmetric relations. An ordered pair of the form of is formed by a relation R specified on a set A with n elements (a, b). We now know that element ‘a’ can be chosen in n different ways, and element ‘b’ can also be chosen in n different ways. This means that in R, we have n2 ordered pairs (a, b). Also, if (a, b) is in R, (b, a) is compelled to be in R for a symmetric relation. As a result, there are 2n(n-1)/2 ordered pairs. We have ordered pairs of the form (a, a) that are also symmetric for a reflexive relation. There are 2n ordered pairs in total. As a result, there are 2n symmetric relations. 2n. 2n(n-1)/2 = 2n(n+1)/2

Important Notes on Symmetric Relations

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

• 2n(n+1)/2 is the number of symmetric relations on a set with the number ‘n’ of elements.

• If and only if (a, b) ∈ R, then (b, a) ∉ R, for all a, b ∈ A, a relation R on a set A is said to be asymmetric.

Conclusion

‘A new set containing all the items present in either of the sets but not in their intersection is defined as a symmetric definition. Also, it is the difference between any two sets.’ Geometry, nature, and shapes are all based on symmetry. It generates patterns that assist us in conceptually organising our reality. When analysing mathematical issues, individuals must perform transformations and use symmetry.