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.
A relation \( R \) is reflexive if for every element \( a \in I \), the pair \( (a,a) \) is in \( R \). Since \( R \) does not contain \( (4,4), (5,5), (6,6), (7,7) \), it is not reflexive.

Step 2: Symmetric Property.
A relation is symmetric if \( (a,b) \in R \) implies \( (b,a) \in R \). The presence of pairs like \( (4,5) \) and \( (5,4) \) (and others) confirms that \( R \) is symmetric.

Step 3: Transitive Property.
A relation is transitive if \( (a,b) \in R \) and \( (b,c) \in R \) implies \( (a,c) \in R \). However, \( (4,5) \in R \) and \( (5,4) \in R \) would require \( (4,4) \in R \) for transitivity. As \( (4,4) \) is not in \( R \), the relation is not transitive.

Step 4: Antisymmetric Property.
A relation is antisymmetric if \( (a,b) \in R \) and \( (b,a) \in R \) implies \( a = b \). For example, \( (4,5) \in R \) and \( (5,4) \in R \), but \( 4 eq 5 \). Therefore, \( R \) is not antisymmetric.

Step 5: Conclusion.
As the relation lacks the transitive property, the conclusion is that options A, C, and D are correct.