JEE Mathematics 2025 Question with Solution

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

Find: The number of such relations.

A reflexive relation on $A$ must contain

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

Since $(1,2)$ and $(2,3)$ are given, transitivity forces

$$(1,3)$$

because

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

So the minimal relation is

$$R_0 = \{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$$

Now the only possible additional reverse pairs are

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

We check which subsets of these can be added while preserving transitivity.

Valid transitive extensions are:

1. No additional pair
2. $+(2,1)$
3. $+(3,2)$
4. $+(3,1)$
5. $+(2,1),(3,1)$
6. $+(3,2),(3,1)$
7. $+(2,1),(3,2),(3,1)$

The pair combination $(2,1)$ and $(3,2)$ together forces $(3,1)$ by transitivity, so that case is already included only in the seventh relation.

Each of these $7$ relations is not symmetric, because at least one of $(1,2), (2,3), (1,3)$ does not have its reverse pair present in full.

Therefore, the number of required relations is $7$.

Given: $A = \{1,2,3\}$ and the relation contains $(1,2)$ and $(2,3)$.

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

Reflexive means all diagonal pairs must be present:

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

Transitivity with the given pairs gives:

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

Hence every valid relation must contain

$$R_0 = \{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$$

Now consider the optional reverse pairs:

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

Check all transitive possibilities:

• Add none: valid.
• Add only $(2,1)$: valid.
• Add only $(3,2)$: valid.
• Add only $(3,1)$: valid.
• Add $(2,1)$ and $(3,1)$: valid.
• Add $(3,2)$ and $(3,1)$: valid.
• Add $(2,1)$ and $(3,2)$: then transitivity requires $(3,1)$, so this is not a separate case.
• Add all three: valid.

Thus the total number of valid relations is

$$7$$

Also, none of these $7$ relations is symmetric, because symmetry would require all reverse pairs of the non-diagonal elements, which does not happen here.

Therefore, the answer is $7$.