A Short Note On Symmetric Relation

Symmetric Relation is derived from a very common word “symmetry”. Symmetry is the type of attribute of a figure or object that makes it the same on both sides of an axis. Similarly, symmetric relations tell some interlinked connection between symmetry attributes and a relation. If there is a relationship established between two or more elements of any set, then that relationship is known as symmetric relation. Asymmetric relation between two elements is always a binary relation. Alongside symmetric relations, there are several more relations that you’ll study in discrete mathematics. They are asymmetric(anti-symmetric), transitive, reflexive, etc. In this informative article, you will understand the fundamental concept of symmetric relations and the formulas involved in symmetric relations to find the numbers of total symmetric relations. 

What are Symmetric Relations?

A binary relation L defined on a set S is said to be symmetric if, for elements p, q ∈ S, we have pLq, that is, (p, q) ∈ L, then we must have qLp, that is, (q, p) ∈ L. A relation defined on a set S is a symmetric relation if and only if it satisfies pLq ⇔ qLp for all elements p, q in S. If there is a single ordered pair in L such that (p, q) ∈ L and (q, p) ∉ L, then L is not a symmetric relation.

A symmetric relation is a binary relation. So, let’s suppose there is a set S which has a binary relation L for the elements available in the set S shares a relation (pLq). For the same elements, there should be a relation (qLp) also coexisting at the same time for the whole relation to be called symmetric relation.   

pLq ⇔ qLp

Both the elements should follow the above rule and should be inside the set S. If a single pair in the symmetric relation L such that two elements p and q belongs to the group of real numbers, but if any of these p or q do not belong to the group of real numbers, then that relations will not be a symmetric relation. 

Symmetric Relation Formula

The symmetric relation formula will tell you the total number of symmetric relations that have been established between n elements of the set, where each member of the set has to be in some kind of relation with the remaining elements in the set from both ways. 

N = 2n(n+1)/2

Here, N denotes the total count of symmetric relations established, and n denotes the count of elements in the set. 

Solved Example

Example 1: Assume L to be a relation on a set S where S = {1, 3, 5} and L = { (1,1), (5,3), (3,5), (3,3)}.  Is L a symmetric relation between elements of set S. 

Solution: As we can see (3,5) ∈ L. For L to be symmetric (5, 3) should be present in the relation L, and luckily (5,3) is available on the relation L.

Therefore, R is a symmetric relation.

Example 2: L is a relation on a set S where S = {p, q, r} and L = {(p, p), (p, q), (p, r), (q, r), (r, p)}. Find out the pair of elements of alphabets that should be in the relation L to make L a symmetric relation for the set S.

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

Because (p, p) belongs to L,

Because (q, p) belongs to L, but (q, p) does not belongs to L

Because (r, p) belongs to L, 

Because (r, q) belongs to L, but (r, q) does not belongs to L

Hence, (q, p)) and (r, q) should belong to L to make L a symmetric relation.

Conclusion

A relation L is symmetric only if (q, p) belongs to L and when (p and q the elements of set S also belong to L. An example of symmetric relation can be L = {(10, 12), (12, 10), (5,6), (7,13), (3, 6), (13, 7} for a set S= {10, 12} and one more pair of elements is subjected similar to this i.e. (13,7) and (7,13).  The final formula or the condition for the symmetric relation that works the best is pLq ⇒ qLp, ∀ p, q ∈ S.  We got to know what is symmetric relation, what are the formulas associated with it.