Antisymmetric Relation

Antisymmetric Relation

"What is an antisymmetric relation?" asked Miss Granger in the geometry class.

Nicole, who had always loved discrete mathematics said, "A relation in which an element 'a' is related to another element 'b' and 'b' is related to 'a' only if a=b."

When you ask someone the question: 'What is an antisymmetric relation?' then Nicole's answer is almost perfect.

In this section, we’re going to look at the antisymmetric relation examples and antisymmetric relations.

Antisymmetric Relation

Lesson Plan

What Is Meant By Antisymmetric Relation? 

Let us first recall the meaning of a relation.

A relation is just a collection of ordered pairs.

The first element n in the ordered pair is related to the second element according to the relation defined.

The set of input numbers is called the domain of the relation, and the set of output numbers is called the range of the relation. 

Let us now understand the meaning of antisymmetric relations.

A relation R on a set A is said to be antisymmetric if there does not exist any pair of distinct elements of A which are related to each other by R.

Mathematically, it is denoted as:

For all a, b \( \in \) A,

If (a,b) \(\in\) R and (b,a) \(\in\) R, then a=b

Equivalently, 

For all a, b \( \in \) A,

If (a,b) \(\in\) R and a \(\neq\) b, then (b,a) \(\in\) R must not hold.


Where Can You Use the Concept of Antisymmetric Relation? 

Let us explore some examples where antisymmetric relations can be used.

Example

Define a relation R on a set X as:

An element \(x\) in X is related to an element \(y\) in X as \(x\) is divisible by \(y\).

That is, \((x,y) \in\) R if and only if \(x\) is divisible by \(y\)

We will determine if R is an antisymmetric relation or not.

Assume \((x,y) \in R\) and \((y,x) \in R\).

This implies \(x\) is divisible by \(y\) and \(y\) is divisible by \(x\)

This is possible only if \(x=y\)

To verify this, let us consider particular values of \(x\) and \(y\)

Suppose \(x=4\) and \(y=2\)

We can say that (4,2) \(\in\) R but (2,4) \(\notin\) R because 4 is divisible by 2 but 2 is not divisible by 4

Hence, R is an antisymmetric relation.

 
important notes to remember
Important Notes
  1. A relation is a collection of ordered pairs.
  2. A relation R on a set A is said to be antisymmetric if there does not exist any pair of distinct elements of A which are related to each other by R.

Solved Examples 

Example 1

 

 

Help Ron determine if the following relation R is antisymmetric defined on set A = {1,3,5,7,8}

R={(5,5),(1,1),(7,7),(3,3),(8,8)}

Solution

There does not exist any pair of distinct elements of A which are related to each other by R.

For every a,b \(\in\) A, (a,b) \(\in\) R and (b,a) \(\in\) R only when a = b

Therefore, R is an antisymmetric relation.

\(\therefore\) R is an antisymmetric relation.
Example 2

 

 

Sam's mother brought chocolate cookies for Sam and his seven friends. 

The condition is that the number of cookies is divisible by the number of children and the number of children is divisible by the number of cookies.

Define a relation and determine if it is antisymmetric.

Example of antisymmetric relation

Solution

The ordered pairs in relation R can be defined as:

  • (Number of cookies, Number of children)
  • (Number of children, Number of cookies)

Both ordered pairs are in relation R

The number of children is 8 including Sam and his seven friends.

Therefore, the number of cookies is divisible by 8, and 8 is divisible by the number of cookies.

The only number which satisfies the condition is 8

This implies the number of cookies is 8

R={(8,8)} which is antisymmetric.

\(\therefore\) R={(8,8)}

Interactive Questions 

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result. 

 
 
 
 
 
Thinking out of the box
Think Tank
  1. Are all antisymmetric relations reflexive?
  2. Are all antisymmetric relations asymmetric?
  3. Is an empty relation antisymmetric?

Let's Summarize

We hope you enjoyed learning about antisymmetric relation with the solved examples and interactive questions. Now you will be able to easily solve questions related to the antisymmetric relation.

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Frequently Asked Questions (FAQs)

1. Can a relation be symmetric and antisymmetric?

Yes, a relation can be symmetric and antisymmetric. 

For example, R = {(1,1),(2,2), (3,3)} is symmetric as well as antisymmteric.

2. How to prove a relation is antisymmetric?

If in a relation R, (a, b) and (b, a) belong to R, then we need to prove that a=b

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