Question

Consider the following relation \( R = \{(4,5), (5,4), (7,6), (6,7)\} \) on set \( I = \{4,5,6,7\} \). Which of the following properties relation \( R \) does not have? (A) Reflexive property
(B) Symmetric property
(C) Transitive property
(D) Antisymmetric property
Choose the correct answer from the options given below:

• A. A, C and D only
• B. A, B and D only
• C. A, B, C and D
• D. B, C and D only

Hint:

A relation is transitive if, for any \( a, b, c \in I \), whenever \( (a,b) \) and \( (b,c) \) are in the relation, \( (a,c) \) must also be in the relation.

Answer 1

The Correct Option is A

Step 1: Reflexive Property.
For the relation \( R \) to be reflexive, it must contain the pair \( (a,a) \) for every element \( a \in I \). Here, the relation does not contain \( (4,4), (5,5), (6,6), (7,7) \), so \( R \) is not reflexive.

Step 2: Symmetric Property.
For the relation to be symmetric, if \( (a,b) \) is in the relation, then \( (b,a) \) must also be in the relation. Here, \( (4,5) \) and \( (5,4) \) are present, and similarly for the other pairs, making \( R \) symmetric.

Step 3: Transitive Property.
For the relation to be transitive, if \( (a,b) \) and \( (b,c) \) are in the relation, then \( (a,c) \) must also be in the relation. However, \( (4,5) \) and \( (5,4) \) should imply \( (4,4) \), but \( (4,4) \) is not in the relation, so \( R \) is not transitive.

Step 4: Antisymmetric Property.
For the relation to be antisymmetric, if \( (a,b) \) and \( (b,a) \) are in the relation, then \( a = b \). In this case, both \( (4,5) \) and \( (5,4) \) are present, but \( 4 \neq 5 \) and \( 5 \neq 4 \), so \( R \) is not antisymmetric.

Step 5: Conclusion.
The relation does not have the transitive property, so the correct answer is (1) A, C and D only.