JEE Mathematics 2023 Question with Solution

Given: Let $A = \{1, 2, 3\}$. The relation contains $(1,2)$ and $(2,3)$. It must be reflexive and transitive, but not symmetric.

Find: The number of such relations.

For the relation to be reflexive, we must have

$$(1,1), (2,2), (3,3) \in R$$

Since $(1,2)$ and $(2,3)$ are in $R$ and the relation is transitive, we must also have

$$(1,2) \text{ and } (2,3) \in R \implies (1,3) \in R$$

So every such relation must contain

$$\{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$$

Now list the relations given in the solution working:

$$R_1 = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$$ $$R_2 = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3), (2,1)\}$$ $$R_3 = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3), (3,2)\}$$

Hence, the total number of such relations is $3$.

Given: The set is $\{1,2,3\}$ and the relation must contain $(1,2)$ and $(2,3)$.

Find: Count relations that are reflexive and transitive but not symmetric.

1. Reflexive condition: include

$$(1,1), (2,2), (3,3)$$

2. Transitive condition: from

$$(1,2) \text{ and } (2,3)$$

we must include

$$(1,3)$$

3. So the compulsory part is

$$\{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$$

4. The extracted solution lists exactly three admissible relations: $R_1, R_2, R_3$.

Therefore, the required number is $3$.