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Antisymmetric relation is related to sets, functions, and other relations. ‘a’ and ’b’ being assumed as different valued components of a set, an antisymmetric relation is a relation where whenever (a, b) is present in a relation then definitely (b, a) is not present unless ‘a’ is equal to ‘b’.Antisymmetric relation is used to display the relation among the components of a set along with symmetric and asymmetric relations in discrete math.
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Key Takeaways: Symmetric Relation, Asymmetric Relation, Antisymmetric Relations, Sets, Functions, Reflexive and Transitive Relations
Antisymmetric Relation
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A Relation ‘R’ is said to be antisymmetric if (a, b) belongs to R then (b, a) should also belong to ‘R’ only if a = b for all a, b belongs to A.
If (a, b) \(\in\) R then (b, a) \(\in\)R only if a=b(a,b)∈A
E.g., A = {1, 2, 3}
Then R = {(1, 2), (2, 2), (2, 1)}
The first term (1, 2) » (a, b) is present in the relation thus (b, a) » (2, 1) should not be present in the relation as a b as per the definition of Antisymmetric Relation. Thus, the given relation ‘R’ is not an antisymmetric relation.
Consider relation R = {(1, 1), (2, 2), (1, 3)}
The first term (1, 1) » (a, b) is present in the relation, it is a self-symmetric component. Here as a = b as per the definition of Antisymmetric Relation. Thus, the given relation ‘R’ is in antisymmetric relation. Other Antisymmetric relations can be written for the same ‘A’ as -
R = {(1, 1), (2, 2), (3, 3)}
R = {(1, 2), (1, 3), (2, 3)}
R = {}
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Symmetric, Asymmetric and Antisymmetric Relations
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Symmetric, asymmetric, and antisymmetric relations are all very closely related. Each of the relations have slight differences from each other. They can be easily confused; thus, it becomes important to know each individual relation thoroughly for quick identification of the relation.
Symmetric Relation
If (a, b) \(\in\) R and (b, a) \(\in\) R (a,b) \(\in\) A
E.g., A = {1, 2, 3, 4}
Then R = {(1, 2), (1, 3), (2, 2), (2, 1), (3, 1)}
The first term (1, 2) » (a, b) is present in the relation thus (b, a) » (2, 1) should also be present in the relation. The second term (1, 3) » (a, b) is present in the relation thus (b, a) » (3, 1) should also be present in the relation. There is also a term (2, 2) where the inverse of the term gives the same term thus it satisfies the condition of symmetrical relation. Thus, the above relation is a symmetric relation.
R = {} also belongs to symmetric relation.
Asymmetric & Antisymmetric Relation
If (a, b) \(\in\) R then (b, a) R(a,b) \(\in\) A
E.g., A = {1, 2, 3}
Then R = {(1, 2), (1, 3), (1, 1)}
The first term (1, 2) » (a, b) is present in the relation thus (b, a) » (2, 1) should not be present in the relation. There is also a term (1, 1) where a = b. Asymmetric relation does not uphold a = b condition instead antisymmetric does. Thus, the given relation ‘R’ is not in asymmetric but in antisymmetric relation.
Consider relation R = {(1, 2), (1, 3), (2, 3)}
The first term (1, 2) » (a, b) is present in the relation, thus (b, a) » (2, 1) is not present in the relation as per asymmetric condition. Here as a b as per the definition of Asymmetric Relation. Thus, satisfying both the conditions, the given relation ‘R’ is an asymmetric relation.
R = {} also belongs to asymmetric relation.
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Things to Remember
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- For an antisymmetric relation, (a, b) cannot be written as (b, a) or a = b either of both conditions should satisfy for an antisymmetric relation to be justified.
- If a relation is not classified as symmetric then it is not necessary that it is antisymmetric.
- A collection of ordered pairs is called a relation.
- A null set can be symmetric, antisymmetric, and asymmetric relation.
- It is important to remember what does not belong to a particular relation for identifying a particular relation.
- Every asymmetric relation is an antisymmetric relation.
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Sample Questions
Ques. Determine if the following relation is symmetric or not. A = {1, 2, 3}. (3 marks)
a) R = {(1, 1), (1, 2), (1, 3), (2,3), (3, 1)}
b) R = {(1, 2), (2, 2), (1, 3)}
c) R = {(1, 3), (1, 2), (3, 3), (3, 1), (2, 1)}
d) R = {(1, 1), (3, 1), (1, 3)}
Ans.
- The component (1, 2) should be having an inverse in the relation i.e. (2, 1) which seems to be missing thus the given relation does not belong to symmetric relationb
- The component (1, 2) and (1, 3) should be having an inverse in the relation i.e. (2, 1) and (3, 1) which seems to be missing as per symmetric relation conditions thus the given relation does not belong to symmetric relationc.
- The relation given has the component and its inverse in correct format as per the condition of symmetric relation, thus, it is a symmetric relation.d.
- The relation given has the component and its inverse in correct format as per the condition of symmetric relation, thus, it is a symmetric relation.
Ques. Which of the following are antisymmetric? (3 marks)
R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
R = {(1, 1), (1, 3), (3, 1)}
R = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}
Ans.
- Here, R is not antisymmetric as (1, 2) ∈ R and (2, 1) ∈ R, but 1 ≠ 2.
- R is not antisymmetric because of (1, 3) ∈ R and (3, 1) ∈ R, however, 1 ≠ 3.
- Here, R is not antisymmetric because of (1, 2) ∈ R and (2, 1) ∈ R, but 1 ≠ 2. Also, (1, 4) ∈ R, and (4, 1) ∈ R, but 1 ≠ 4.
Ques. Given a set {1, 2, 3, 4}, how is the following relation R antisymmetric? (2 marks)
R= {(1, 2), (2, 3), (3, 4)}
Ans. The component (1, 2) should be not having an inverse in the relation i.e. (2, 1) which holds true.
If (a, b) \(\in\)R then (b, a) \(\in\)R only if a=b∀(a,b)∈A
There is no a = b. Thus, the given relation is an antisymmetric relation.
Ques. Determine if the following is antisymmetric or not? (2 marks)
R = {(1, 2), (1, 3), (3, 1), (1, 1), (3, 3), (3, 2), (1, 4), (4, 2), (3, 4)}
Ans.
First step is to find 2 members in the relation such that (a, b) ∈R (a, b) ∈R and (b, a) ∈R (b, a) ∈R. If no such pair exists, then your relation is antisymmetric. If any such pair exists in your relation and a ≠ ba ≠ b then the relation is not antisymmetric, otherwise it is anti-symmetric.
Ques. What are the conditions for Symmetrical, Asymmetrical, and Antisymmetric relations? (3 marks)
Ans. Symmetric relations
If (a, b) \(\in\)R and (b, a) \(\in\)R ∀(a,b)∈A
Asymmetric relations
If (a, b) \(\in\)R then (b, a) R∀(a,b)∈A
Antisymmetric relations
If (a, b) \(\in\)R then (b, a) \(\in\)R only if a=b∀(a,b)∈A
Ques. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive. (5 marks)
Ans. Let A = {1, 2, 3}
A relation R on A is defined as R = {(1,2), (2, 1)}
It is seen that (1, 1), (2, 1), (3, 3) R
Thus, R is not reflexive.
Now, as (1, 2) \(\in\)R and (2, 1)\(\in\)R, the R is symmetric.
Now, (1, 2) and (2, 1) \(\in\)R
However,
(1, 1) R
Thus, R is not Transitive
Hence, R is symmetric but neither reflexive nor transitive.
Ques. Give an example of a relation which is; (5 marks)
a) Symmetric but neither reflexive nor transitive.
b) Transitive but neither reflexive nor symmetric.
Ans.
- Let A = {5, 6, 7}
Thus, we need to define a relation R on A as R = {(5, 6), (6,5)}
Relation R is not reflexive thus (5, 5), (6, 6), (7, 7) R
Now as (5, 6) \(\in\)R and (6, 5) \(\in\)R , R is symmetric.
(5, 6), (6, 5) \(\in\)R but (5, 5) R
Thus, R is not transitive. Symmetric but neither reflexive nor transitive.
- Consider a relation R where R is defined as R = {(a, b): a < b}
For any a \(\in\)R, we have (a, a) R since a cannot be strictly less than a itself. In fact a = a, thus, R is not reflective.
Now, (1, 2) \(\in\)R (as 1 < 2)
But 2 is not less than 1
Thus, (2, 1) R, R is not symmetric.
Now, let (a, b), (b, c) \(\in\)R
= a < b and b < c
= a < c
= (a, c) \(\in\)R, R is transitive, Transitive but neither reflexive nor symmetric.
Ques. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. (5 marks)
Ans. Let A = {1, 2, 3, 4, 5, 6}
A relation R is defined on set A as
R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}
We can find (a, a) \(\in\)R, where a A.
For instance,
(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) R
Thus, R is not reflexive.
It is observed that (1, 2) \(\in\)R but (2, 1) R.
Thus, R is not Symmetric.
Now, (1, 2), (2, 3) \(\in\)R
But,
(1, 3) R.
Thus, R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive
Ques. Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric. (5 marks)
Ans. R = {(a, b): a ≤ b}
Clearly (a, a) \(\in\)R as a = a.
Thus, R is reflexive.
Now (2, 4) \(\in\)R (as 2 < 4)
But (4, 2) R as 4 is greater than 2.
Thus, R is not symmetric.
Now, let (a, b), (b, c) \(\in\)R
Then,
a≤b and b≤c
= a≤c
= (a, c) \(\in\)R
Thus, R is transitive.
Hence R is reflexive and transitive but not symmetric.
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