Suppose X = {1, 2, 3, 4} and R is a relation on X. If R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}, then which one of the following is correct?
R is symmetric and transitive, but not reflexive
Understanding Relation Properties
Let's analyze the given relation R on the set X to determine its properties: reflexive, symmetric, and transitive. The set is X = {1, 2, 3, 4}, and the relation R is given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}.
Checking for Reflexive Property in Relation R
A relation R on a set X is said to be reflexive if for every element \(x \in X\), the pair \((x, x)\) belongs to R. In simpler terms, every element must be related to itself.
For the given set X = {1, 2, 3, 4}, for R to be reflexive, the following pairs must be present in R:
(1, 1)
(2, 2)
(3, 3)
(4, 4)
Let's look at the given relation R:
R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}
We can see that (1, 1), (2, 2), and (3, 3) are present in R. However, the pair (4, 4) is missing from R. Since there is an element in X (which is 4) that is not related to itself, the relation R is not reflexive.
Conclusion for Reflexivity: R is not reflexive.
Checking for Symmetric Property in Relation R
A relation R on a set X is said to be symmetric if for every pair \((x, y)\) in R, the reversed pair \((y, x)\) also belongs to R. This means that if x is related to y, then y must also be related to x.
Let's examine each pair \((x, y)\) in R and check if \((y, x)\) is also in R:
(1, 1) is in R. Is (1, 1) in R? Yes.
(2, 2) is in R. Is (2, 2) in R? Yes.
(3, 3) is in R. Is (3, 3) in R? Yes.
(1, 2) is in R. Is (2, 1) in R? Yes, (2, 1) is in R.
(2, 1) is in R. Is (1, 2) in R? Yes, (1, 2) is in R.
(2, 3) is in R. Is (3, 2) in R? Yes, (3, 2) is in R.
(3, 2) is in R. Is (2, 3) in R? Yes, (2, 3) is in R.
(1, 3) is in R. Is (3, 1) in R? Yes, (3, 1) is in R.
(3, 1) is in R. Is (1, 3) in R? Yes, (1, 3) is in R.
For every pair \((x, y)\) in R, the corresponding pair \((y, x)\) is also found in R.
Conclusion for Symmetry: R is symmetric.
Checking for Transitive Property in Relation R
A relation R on a set X is said to be transitive if for every pair \((x, y)\) in R and every pair \((y, z)\) in R, the pair \((x, z)\) also belongs to R. This means if x is related to y, and y is related to z, then x must also be related to z.
We need to check all possible combinations of pairs \((x, y) \in R\) and \((y, z) \in R\). If such pairs exist, we must verify that \((x, z) \in R\).
Let's list some examples:
Take (1, 2) \(\in\) R and (2, 1) \(\in\) R. Here, \(x=1, y=2, z=1\). We need to check if (1, 1) \(\in\) R. Yes, (1, 1) is in R.
Take (1, 2) \(\in\) R and (2, 3) \(\in\) R. Here, \(x=1, y=2, z=3\). We need to check if (1, 3) \(\in\) R. Yes, (1, 3) is in R.
Take (2, 3) \(\in\) R and (3, 1) \(\in\) R. Here, \(x=2, y=3, z=1\). We need to check if (2, 1) \(\in\) R. Yes, (2, 1) is in R.
Take (3, 1) \(\in\) R and (1, 2) \(\in\) R. Here, \(x=3, y=1, z=2\). We need to check if (3, 2) \(\in\) R. Yes, (3, 2) is in R.
The elements involved in the pairs in R are only {1, 2, 3}. The element 4 is not involved in any pairs in R. If (x, y) \(\in\) R and (y, z) \(\in\) R, then x, y, and z must be from the set {1, 2, 3}. The given relation R restricted to the set {1, 2, 3} contains all possible pairs between elements from {1, 2, 3}, i.e., it is the universal relation on {1, 2, 3}. The universal relation on any set is always transitive. Thus, if (x, y) \(\in\) R and (y, z) \(\in\) R, then x, y, z \(\in\) {1, 2, 3}, and (x, z) must be in R because all pairs of elements from {1, 2, 3} are in R.
After checking all combinations (which can be extensive but follows the pattern seen above), we find that the transitive property holds for R.
Conclusion for Transitivity: R is transitive.
Summary of Relation R Properties
Based on our analysis:
R is not reflexive (missing (4, 4)).
R is symmetric.
R is transitive.
Therefore, the relation R is symmetric and transitive, but not reflexive.
Conclusion on Relation R Properties
The relation R on set X = {1, 2, 3, 4} is found to be symmetric and transitive, but not reflexive. This matches one of the given options describing the properties of the relation R.
| Property | Definition | Check for Relation R | Conclusion |
|---|---|---|---|
| Reflexive | For all \(x \in X\), \((x, x) \in R\) | (4, 4) \(\notin\) R (since 4 \(\in\) X) | Not Reflexive |
| Symmetric | For all \((x, y) \in R\), \((y, x) \in R\) | Checked all pairs; holds true for R | Symmetric |
| Transitive | For all \((x, y) \in R\) and \((y, z) \in R\), \((x, z) \in R\) | Checked combinations; holds true for R | Transitive |
Revision Table: Key Relation Definitions
| Property | Condition |
|---|---|
| Reflexive | \((a, a) \in R\) for all \(a \in\) set A |
| Symmetric | If \((a, b) \in R\), then \((b, a) \in R\) for all \(a, b \in\) set A |
| Transitive | If \((a, b) \in R\) and \((b, c) \in R\), then \((a, c) \in R\) for all \(a, b, c \in\) set A |
Additional Information on Relation Properties
Relations are fundamental concepts in mathematics and computer science, used to describe relationships between elements of sets. Understanding the properties like reflexive, symmetric, and transitive helps classify different types of relations.
An equivalence relation is a relation that is simultaneously reflexive, symmetric, and transitive. Equivalence relations partition a set into disjoint equivalence classes.
A partial order relation is a relation that is reflexive, antisymmetric, and transitive. Antisymmetric means if \((a, b) \in R\) and \((b, a) \in R\), then \(a=b\).
The relation R discussed in this problem is an example of a relation that satisfies symmetry and transitivity but fails reflexivity on the entire set X due to the element 4.
When testing these properties, it is crucial to consider all elements in the set X, not just those that appear in the ordered pairs listed in the relation R.
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