Relations: Analyzing Reflexive, Transitive, and Non-Symmetric Properties

To identify which of the given relations on the set {1, 2, 3, 4} are reflexive and transitive but NOT symmetric, we need to carefully examine each relation (R1, R2, R3, R4) based on the definitions of these fundamental properties.

Relation Properties Definitions

• Reflexive Relation: A relation R on a set A is considered reflexive if for every element \(a\) belonging to set A, the ordered pair \((a, a)\) is present in R. For the given set {1, 2, 3, 4}, a reflexive relation must include the pairs \((1, 1), (2, 2), (3, 3), \text{ and } (4, 4)\).
• Symmetric Relation: A relation R on a set A is symmetric if, whenever an ordered pair \((a, b)\) is in R, its reverse pair \((b, a)\) is also in R. That is, if \((a, b) \in R\) implies \((b, a) \in R\).
• Transitive Relation: A relation R on a set A is transitive if, whenever we have two ordered pairs \((a, b)\) and \((b, c)\) in R, it implies that the ordered pair \((a, c)\) must also be in R. That is, if \((a, b) \in R\) and \((b, c) \in R\) implies \((a, c) \in R\).

Analyzing Relation R1

The relation R1 is given as: R1 = \(\{(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)\}\).

• Reflexive Test:
  • The pairs \((1, 1), (2, 2), (3, 3), \text{ and } (4, 4)\) are all present in R1.
  • Therefore, R1 is reflexive.
• Symmetric Test:
  • Let's check if for every \((a, b) \in R1\), \((b, a) \in R1\).
    • \((1, 2) \in R1\) and \((2, 1) \in R1\).
    • \((1, 4) \in R1\) and \((4, 1) \in R1\).
    • All other pairs like \((1,1), (2,2)\) are trivially symmetric.
  • Therefore, R1 is symmetric. Since the requirement is "NOT symmetric", R1 does not meet this condition.
• Transitive Test:
  • We check if for every \((a, b) \in R1\) and \((b, c) \in R1\), then \((a, c) \in R1\).
    • Consider the pairs \((2, 1) \in R1\) and \((1, 4) \in R1\). For transitivity to hold, \((2, 4)\) must be in R1.
    • However, \((2, 4)\) is NOT found in R1.
  • Therefore, R1 is NOT transitive.

Conclusion for R1: R1 is Reflexive and Symmetric, but NOT Transitive. It does not meet the specified criteria.

Analyzing Relation R2

The relation R2 is given as: R2 = \(\{(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)\}\).

• Reflexive Test:
  • The pairs \((1, 1), (2, 2), (3, 3), \text{ and } (4, 4)\) are NOT present in R2. For instance, \((1, 1)\) is missing.
  • Therefore, R2 is NOT reflexive. Since the requirement is "reflexive", R2 does not meet this condition.
• Symmetric Test:
  • We check if for every \((a, b) \in R2\), \((b, a) \in R2\).
    • Consider \((2, 1) \in R2\). For symmetry, \((1, 2)\) must be in R2.
    • However, \((1, 2)\) is NOT in R2.
  • Therefore, R2 is NOT symmetric. This part satisfies the "NOT symmetric" condition.
• Transitive Test:
  • Let's check various combinations for transitivity:
    • If \((3, 2) \in R2\) and \((2, 1) \in R2\), then \((3, 1)\) must be in R2. Yes, it is present.
    • If \((4, 3) \in R2\) and \((3, 1) \in R2\), then \((4, 1)\) must be in R2. Yes, it is present.
    • If \((4, 3) \in R2\) and \((3, 2) \in R2\), then \((4, 2)\) must be in R2. Yes, it is present.
    • Upon thoroughly checking all possible pairs, R2 satisfies the transitivity condition.
  • Therefore, R2 is transitive.

Conclusion for R2: R2 is NOT Reflexive, NOT Symmetric, and Transitive. It does not meet all the specified criteria.

Analyzing Relation R3

The relation R3 is given as: R3 = \(\{(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)\}\).

• Reflexive Test:
  • The pairs \((1, 1), (2, 2), (3, 3), \text{ and } (4, 4)\) are all present in R3.
  • Therefore, R3 is reflexive. This satisfies the condition.
• Symmetric Test:
  • We check if for every \((a, b) \in R3\), \((b, a) \in R3\).
    • Consider \((1, 2) \in R3\). For symmetry, \((2, 1)\) must be in R3.
    • However, \((2, 1)\) is NOT in R3.
    • Similarly, \((1, 3), (1, 4), (2, 3), (2, 4), \text{ and } (3, 4)\) are in R3, but their reverse pairs are not.
  • Therefore, R3 is NOT symmetric. This satisfies the condition.
• Transitive Test:
  • R3 represents a "less than or equal to" relation on the set elements. If \((a, b) \in R3\) and \((b, c) \in R3\), it implies \(a \le b\) and \(b \le c\). By the property of order, this means \(a \le c\), so \((a, c)\) must be in R3.
    • For example, \((1, 2) \in R3\) and \((2, 3) \in R3\). For transitivity, \((1, 3)\) must be in R3. Yes, it is.
    • Another example: \((2, 3) \in R3\) and \((3, 4) \in R3\). For transitivity, \((2, 4)\) must be in R3. Yes, it is.
    • Upon comprehensive verification, all transitive conditions hold for R3.
  • Therefore, R3 is transitive. This satisfies the condition.

Conclusion for R3: R3 is Reflexive, NOT Symmetric, and Transitive. This relation meets all the specified criteria.

Analyzing Relation R4

The relation R4 is given as: R4 = \(\{(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)\}\).

• Reflexive Test:
  • The pairs \((1, 1), (2, 2), (3, 3), \text{ and } (4, 4)\) are expected for reflexivity. However, \((3, 3)\) is NOT present in R4.
  • Therefore, R4 is NOT reflexive. Since the requirement is "reflexive", R4 does not meet this condition.
• Symmetric Test:
  • We check if for every \((a, b) \in R4\), \((b, a) \in R4\).
    • \((1, 2) \in R4\) and \((2, 1) \in R4\).
    • However, \((3, 4) \in R4\), but \((4, 3)\) is NOT in R4.
  • Therefore, R4 is NOT symmetric. This part satisfies the "NOT symmetric" condition.
• Transitive Test:
  • We check if for every \((a, b) \in R4\) and \((b, c) \in R4\), then \((a, c) \in R4\).
    • Consider the pairs \((4, 1) \in R4\) and \((1, 2) \in R4\). For transitivity, \((4, 2)\) must be in R4.
    • However, \((4, 2)\) is NOT found in R4.
  • Therefore, R4 is NOT transitive.

Conclusion for R4: R4 is NOT Reflexive, NOT Symmetric, and NOT Transitive. It does not meet the specified criteria.

Summary of Relation Properties

Final Conclusion

Based on the detailed analysis of each relation's properties, only relation R3 fulfills all the specified conditions: it is reflexive, it is transitive, and it is NOT symmetric.