Symmetric and Antisymmetric Relation

Introduction

Let us first understand,

What does Symmetric mean?

In maths, It’s the relationship between two or more elements such that if the 1st element is related to the 2nd then the 2nd element is also related to 1st element in a similar manner. For example,

Symmetric Property

The relation \(a = b\) is symmetric, but \(a>b\) is not.

In other words, we can say symmetric property is something where one side is a mirror image or reflection of the other. Or simply we can say any image or shape that can be divided into identical halves is called symmetrical and each of the divided parts is in symmetrical relationship to each other. Otherwise, it would be antisymmetric relation.

In the above diagram, we can see different types of symmetry

Two objects are symmetrical when they have the same size and shape but different orientations.

Let’s consider some real-life examples of symmetric property.

Also read:

• What is a Venn Diagram?

Imagine a sun, raindrops, rainbow. Their structure is such that we can divide them into equal and identical parts when we run a line through them Hence it is a symmetric relation. But if we take the distribution of chocolates to students with the top 3 students getting more than the others, it is an antisymmetric relation.

Symmetric Relation

Any relation R in a set A is said to be symmetric if (a, b) ∈ R. This implies that

\[(b, a) ∈ R\]

In other words, a relation R in a set A is said to be in a symmetric relationship only if every value of a,b ∈ A, (a, b) ∈ R then it should be (b, a) ∈ R.

Suppose R is a relation in a set A where A = {1,2,3} and R contains another pair R = {(1,1), (1,2), (1,3), (2,3), (3,1)}

Here let us check if this relation is symmetric or not.

(1,2) ∈ R but no pair is there which contains (2,1)

We have seen above that for symmetry relation if (a, b) ∈ R then (b, a) must ∈ R.

So, for R = {(1,1), (1,2), (1,3), (2,3), (3,1)} in symmetry relation we must have (2,1), (3,2). Then only we can say that the above relation is in symmetric relation.

i.e. R = {(1,1), (1,2), (1,3), (2,3), (3,1), (2,1), (3,2)}

Let us take another example,

Suppose R is a relation in a set A = {set of lines}

and R = {(L₁, L₂): L₁ is parallel to L_2}

Let’s understand whether this is a symmetry relation or not.

If R = {(L₁, L₂₎}

In all such pairs where L₁ is parallel to L₂ then it implies L₂ is also parallel to L₁.

This means R = {(L₁, L₂), (L_2, L₁)}

It means this type of relationship is a symmetric relation.

Symmetric Relation Example

Example 1

 

 

If A = {a,b,c} so A*A that is matrix representation of the subset product would be

a	aa	ab	ac
b	ba	bb	bc
c	ca	cb	cc

Which of the below are Symmetric Relations?

1. {(a, b), (b, a)}
2. {(b, c), (c, b), (b, b), (c, c)}
3. {(a, a), (b, b), (c, c)}
4. ø
5. A*A
6. {(a, b), (b, c), (a, c)}

Solution:

1. This is a Symmetric relation as when we flip a, b we get b, a which are in set A and in a relationship R. Here the condition for symmetry is satisfied.
2. In this case (b, c) and (c, b) are symmetric to each other. Further, the (b, b) is symmetric to itself even if we flip it. The same is the case with (c, c), (b, b) and (c, c) are also called diagonal or reflexive pair. Hence it is also in a Symmetric relation.
3. Referring to the above example No. 2 as the (a, a), (b, b), and (c, c) are diagonal and reflexive pairs in the above product matrix, these are symmetric to itself. Hence it is also a symmetric relationship.
4. This is no symmetry as (a, b) does not belong to ø.
5. A*A is a cartesian product. As the cartesian product shown in the above Matrix has all the symmetric. Hence this is a symmetric relationship.
6. In this example the first element we have is (a,b) then the symmetry of this is (b, a) which is not present in this relationship, hence it is not a symmetric relationship.

Example 2

 

 

A relation R is defined on the set Z by “a R b if a – b is divisible by 7” for a, b ∈ Z. Examine if R is a symmetric relation on Z.

Solution:

Let a, b ∈ Z, and a R b hold. Then a – b is divisible by 7 and therefore b – a is divisible by 7.

Thus, a R b ⇒ b R a and therefore R is symmetric.

Example 3

 

 

 A relation R is defined on the set Z (set of all integers) by “aRb if and only if 2a + 3b is divisible by 5”, for all a, b ∈ Z. Examine if R is a symmetric relation on Z.

Solution:

Let a, b ∈ Z and aRb holds i.e., 2a + 3a = 5a, which is divisible by 5. Now, 2a + 3a = 5a – 2a + 5b – 3b = 5(a + b) – (2a + 3b) is also divisible by 5.

Therefore, aRa holds for all a in Z i.e. R is reflexive.

Example 4

 

 

Let R be a relation on T, defined by R = {(a, b): a, b ∈ T and a – b ∈ Z}. Show that R is Symmetric relation.

Solution:

Given R = {(a, b): a, b ∈ T, and a – b ∈ Z}.

Let ab ∈ R ⇒ (a – b) ∈ Z, i.e. (a – b) is an integer.

⇒ - (a – b) is an integer

⇒ (b – a) is an integer

⇒ (b, a) ∈ R

Thus, (a, b) ∈ R ⇒ (b, a) ∈ R, Therefore, R is symmetric.

Example 5

 

 

Let n be given fixed positive integer.

 Let R = {(a, a): a, b ∈ Z and (a – b) is divisible by n}. Show that R is a symmetric relation.

Solution:

Given R = {(a, b): a, b ∈ Z, and (a – b) is divisible by n}. Let ab ∈ R. Then,

ab ∈ R ⇒ (a – b) is divisible by n

⇒ - (a – b) is divisible by n

⇒ (b – a) is divisible by n

⇒ (b, a) ∈ R

Thus, (a, b) ∈ R ⇒ (b, a) ∈ R

Therefore, R is a symmetric relation on set Z.

How to prove a relation is Symmetric

Symmetric Proof

Let \(a, b ∈ Z\) (Z is an integer) such that \((a, b) ∈ R\)

So, a-b is divisible by 3.

Now \(a-b = 3K\) for some integer K

So now how \(a-b\) is related to \(b-a i.e. b – a = - (a-b)\) [ Using Algebraic expression]

Next, \(b-a = - (a-b) = -3K = 3(-K)\)

Which is divisible by 3

So, \((b, a) ∈ R\)

Antisymmetric Relation

Given a relation R on a set A we say that R is antisymmetric if and only if for all (a, b) ∈ R where a ≠ b we must have (b, a) ∉ R.

This means the flipped ordered pair i.e. (b, a) can not be in relation if (a,b) is in a relationship. This is called Antisymmetric Relation.

Example

 

 

Let’s say we have a set of ordered pairs where A = {1,3,7}. Figure out whether the given relation is an antisymmetric relation or not.

\[R_1 =\text{ {(1,3), (3,7), (7,1)}}\]

So, in \(R_1\) above if we flip (a, b) we get (3,1), (7,3), (1,7) which is not in a relationship of \(R_1\)_. We can say that in the above 3 possible ordered pairs cases none of their symmetric couples are into relation, hence this relationship is an Antisymmetric Relation.

Summary

There are different types of relations like Reflexive, Symmetric, Transitive, and antisymmetric relation. In this article, we have focused on Symmetric and Antisymmetric Relations.

A relation R in a set A is said to be in a symmetric relation only if every value of \(a,b ∈ A, (a, b) ∈ R\) then it should be \((b, a) ∈ R.\)

Given a relation R on a set A we say that R is antisymmetric if and only if for all \((a, b) ∈ R\) where a ≠ b we must have \((b, a) ∉ R.\)

We also discussed “how to prove a relation is symmetric” and symmetric relation example as well as antisymmetric relation example.

Written by Rashi Murarka

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