The relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on a set A = {1, 2, 3} is
reflexive transitive but not symmetric
Understanding Relations and Their Properties
We are given a set A = {1, 2, 3} and a relation R defined on A as R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. We need to determine if this relation R is reflexive, symmetric, and/or transitive.
Checking for Reflexive Property of the Relation
A relation R on a set A is said to be reflexive if for every element 'a' belonging to set A, the ordered pair (a, a) is present in the relation R. In mathematical notation, this is:
\(\forall a \in A, (a, a) \in R\)
Our set A is {1, 2, 3}. We need to check if the pairs (1, 1), (2, 2), and (3, 3) are in the relation R.
- Is (1, 1) in R? Yes, (1, 1) is in R.
- Is (2, 2) in R? Yes, (2, 2) is in R.
- Is (3, 3) in R? Yes, (3, 3) is in R.
Since all required pairs (a, a) for every element 'a' in A are present in R, the relation R is reflexive.
Checking for Symmetric Property of the Relation
A relation R on a set A is said to be symmetric if for every ordered pair (a, b) belonging to R, the reversed pair (b, a) is also present in R. In mathematical notation, this is:
\(\forall (a, b) \in R, \text{ if } (a, b) \in R \text{ then } (b, a) \in R\)
We need to check each pair (a, b) in R and see if (b, a) is also there.
- Consider (1, 2) \(\in\) R. Is (2, 1) \(\in\) R? Looking at the list of pairs in R, (2, 1) is not present.
Since we found at least one pair (1, 2) in R for which the corresponding reversed pair (2, 1) is not in R, the relation R is not symmetric.
We could also check others:
- Consider (2, 3) \(\in\) R. Is (3, 2) \(\in\) R? No, (3, 2) is not in R.
- Consider (1, 3) \(\in\) R. Is (3, 1) \(\in\) R? No, (3, 1) is not in R.
The relation R fails the symmetry test because pairs like (1, 2), (2, 3), and (1, 3) are in R, but their reverses are not.
Checking for Transitive Property of the Relation
A relation R on a set A is said to be transitive if for every three elements a, b, and c in A, whenever the ordered pair (a, b) is in R and the ordered pair (b, c) is in R, then the ordered pair (a, c) must also be in R. In mathematical notation, this is:
\(\forall a, b, c \in A, \text{ if } (a, b) \in R \text{ and } (b, c) \in R, \text{ then } (a, c) \in R\)
We need to examine all possible chains (a, b) and (b, c) where both pairs are in R, and then check if (a, c) is also in R. Let's list such chains and their implications:
| (a, b) \(\in\) R | (b, c) \(\in\) R | Implied (a, c) | Is (a, c) \(\in\) R? | Transitivity holds for this chain? |
|---|---|---|---|---|
| (1, 1) | (1, 1) | (1, 1) | Yes | Yes |
| (1, 1) | (1, 2) | (1, 2) | Yes | Yes |
| (1, 1) | (1, 3) | (1, 3) | Yes | Yes |
| (1, 2) | (2, 2) | (1, 2) | Yes | Yes |
| (1, 2) | (2, 3) | (1, 3) | Yes | Yes |
| (1, 3) | (3, 3) | (1, 3) | Yes | Yes |
| (2, 2) | (2, 2) | (2, 2) | Yes | Yes |
| (2, 2) | (2, 3) | (2, 3) | Yes | Yes |
| (2, 3) | (3, 3) | (2, 3) | Yes | Yes |
| (3, 3) | (3, 3) | (3, 3) | Yes | Yes |
We have checked all pairs (a, b) and (b, c) that are in R. For every such chain, the required pair (a, c) is also found to be in R. Therefore, the relation R is transitive.
Conclusion: Properties of the Relation
Based on our analysis:
- The relation R is reflexive.
- The relation R is not symmetric.
- The relation R is transitive.
Thus, the relation R is reflexive and transitive but not symmetric.
Revision Table: Relation Properties Summary
| Property | Definition | Is R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on A = {1, 2, 3} ? | Reason |
|---|---|---|---|
| Reflexive | \(\forall a \in A, (a, a) \in R\) | Yes | (1,1), (2,2), (3,3) are all in R. |
| Symmetric | \(\forall (a, b) \in R, (b, a) \in R\) | No | (1,2) \(\in\) R but (2,1) \(\notin\) R. |
| Transitive | \(\forall (a, b) \in R, (b, c) \in R \implies (a, c) \in R\) | Yes | All chains (a,b) and (b,c) in R imply (a,c) is in R (e.g., (1,2) \(\in\) R, (2,3) \(\in\) R \(\implies\) (1,3) \(\in\) R). |
Additional Information: Types of Relations
Relations are fundamental concepts in set theory and discrete mathematics. Understanding their properties is crucial.
- Equivalence Relation: A relation that is reflexive, symmetric, and transitive. Example: Equality (=) on a set of numbers.
- Partial Order Relation: A relation that is reflexive, antisymmetric, and transitive. Antisymmetric means if (a, b) and (b, a) are in R, then a = b. Example: 'less than or equal to' (\(\le\)) on a set of numbers.
- Strict Partial Order Relation: A relation that is irreflexive, antisymmetric, and transitive. Irreflexive means \(\forall a \in A, (a, a) \notin R\). Example: 'less than' (\(<\)) on a set of numbers.
The relation R in this question is an example of a relation that has some properties but not all, illustrating that these properties are independent (except for specific combinations like equivalence or partial order).
Important Questions from Types of Relations
A Relation in R is defined as R = {(a, b) : a ≤ b2} is ________.
Let S = {1, 2, 3, ...}, A relation R on S × S is defined by xRy if log ax > log ay when a \(\rm = \frac 1 2.\) Then the relation is:
- Let A be {I, m, n}. Let the relation R be {}. Which of the following statements about R is true?
- Which of the following relations is symmetric but neither reflexive nor transitive for a set A= {a, b, c}?
Consider the following relations on the set {1, 2, 3, 4}:
R1 = {(1, 1),(1, 2), (1, 4),(2, 1), (2, 2), (3, 3),(4, 1), (4, 4)}
R2 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
R3 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3 ), (2, 4), (3, 3), (3, 4), (4, 4)}
R4 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
Which of these relations are reflexive and transitive but NOT symmetric?
