Symmetry Relation

Symmetry relation :definition

Symmetry relation is a type of binary relation. According to theory, symmetry relation is the relation in which if one element is related to another element; then another element will also be related to the 1st one.

For example: In equation A, if x is only related to y, then y is also related to x for every y.

In a set, a relation is said to be symmetric only if (x,y) belongs to R, then (y, x) for every a,b belongs to A. in discrete mathematics, relation refers to showing the relationship between two sets.

Symmetric relation example

●   On a set, A ” is equal to” is a symmetric relation defined as if an element x is equal to y, then y is equal to x.

equation: xRy => x=y =>y=x.

●   ” is comparable to” is another type of symmetric relation which is defined on a set as” a” is comparable to “b” if only “b” is comparable to. “

●   ” is a biological sibling” is also a type of symmetric relation. If one character, P, is a biological sibling of another character, Q, then Q is also a biological character.

Asymmetric and Antisymmetric relation and symmetric relation 

1. Asymmetric Relations: Asymmetric relation is the opposite of symmetric relation. A relation is said to be asymmetric if (a, b) belongs to R, then (b, a) does not belong to R.

2.  Antisymmetric relation: a relation on a set A is antisymmetric relation only when if aRb and bRa then a=b for all a,b belongs to A. For understanding it in a better way, let’s consider an example:

Given that set A has three elements 1,2, and 3. R={(1,2)(2,2)(2,1)} . Then state whether the equation is Antisymmetric or not?

The answer is no; the relation is not antisymmetric because a relationship can be antisymmetric only if (x, y) belongs to R, then (y, x) belongs to R only when x=y and the given relation does not fulfil this condition.

3. Symmetric relation: Symmetry relation is a type of binary relation. According to theory, symmetry relation is the relation in which if one element is related to another element; then another element will also be related to the 1st one.

Formula to calculate symmetric relation :

To calculate the number of symmetric relations with the ‘n’ element on a set, the Formula is :

N= 2n(n+1)/2

Where N stands for the number of symmetric relations and n stands for the number of elements.

Important points about symmetric relation :

●   Symmetric relation is a binary relation that is said to be symmetric if and only if elements p, q ∈ A, we have pRq, that is, (p,q) ∈ R, and then we must have qRp is, (q, p) ∈ R.

●   The Formula to calculate the number of symmetric relations with ” n ” number of elements is 2n(n+1)/2

●   A relation is said to be asymmetric only (x, y ) ∈ R, then (y,x) ∉ R, for all x, y ∈ A

●   A relation on set A is an antisymmetric relation if (a, b) ∈ R, then (b, a ) ∈ only when a=b for all a, b ∈ A.

Let us know what Relation is? 

In discrete mathematics, relation refers to showing the relationship between two sets. There are mainly eight types of relations. These are :

1.   Empty relation: empty relation, also known as void relation, is one in which there is no relationship between any element of a set.

2.    Identity relation: identity relation, also known as identity transformation, is the function on set A, which relates every element of the set with itself.

3.    Symmetric relation: symmetric relation is a binary relation that is said to be symmetric if and only if elements p, q ∈ A, we have pRq, that is, (p,q) ∈ R, and then we must have qRp is, (q, p) ∈ R.

4.    Universal relation: relation of any given set can be a universal relation only if every set component is related to the set itself. 

5.    Equivalence relation: a relation can be equivalence only when it is transitive, symmetric, and reflexive.

6.    Inverse relation: an inverse relation is obtained by interchanging the characters of the given set.

7.    Reflexive relation: sometimes, students get confused between reflexive and identity relations. To clear your doubts, you must understand that reflexive relation is the relation that relates every element either with itself or maybe it can also relate with an element.

8.   Transitive relations: transitive is also a type of binary relation that is defined such that if the first element of a set is related to the second element, then the second element will be related to the third element, then the third element will also be related to the first element.

Conclusion

Relations are the ways that show the relationship between any two elements. There are numerous types of relations; among them, one is a symmetric relation. Symmetric relation is a type of binary relation. According to set theory, symmetry relation is the relation in which if one element is related to another element; then another element will also be related to the 1st one. Some of the symmetric relations are” is equal to,” ” is a biological sibling,” and ” is comparable to’. Asymmetric and Antisymmetric relations are also other types of relations. Asymmetric relations are just opposite to symmetric relations.