JAMIA 2026 — Mathematics PYQ

Step 1: Understand the Condition for Reflexivity

A relation $R$ on a set $A$ is called reflexive if every element in $A$ is related to itself.

Mathematically, for every element $a\in A$, the ordered pair $(a,a)$ must belong to the relation $R$.

Given the set $A=\{1,2,3,4\}$, the total number of elements is:

$$n=4$$

Step 2: Breakdown the Cartesian Product ($A\times A$)

The total number of ordered pairs in the Cartesian product $A\times A$ is given by $n\times n=n^2$.

For our set, $4^2=16$ total ordered pairs. These $16$ pairs can be divided into two separate types:

1. Diagonal Elements (Reflexive pairs): Pairs where the first element equals the second element, i.e., $(a,a)$.

   Examples: $\{(1,1),(2,2),(3,3),(4,4)\}$

   • Total diagonal elements $=n=4$

2. Non-Diagonal Elements: Pairs where the first element does not equal the second element, i.e., $(a,b)$ where $a\ne b$.

   Examples: $(1,2),(2,1),(3,4)$, etc.

   • Total non-diagonal elements $=n^2-n=16-4=12$

Step 3: Analyze the Choices for Creating a Reflexive Relation

When forming a reflexive relation, we must make a binary choice (either include or exclude) for each ordered pair:

• For the diagonal elements, there is only 1 choice for each pair: they must be included to keep the relation reflexive.

• For the non-diagonal elements, there are 2 choices for each pair: they can either be included or excluded from the relation.

Using the fundamental counting principle:

$$\text{Total Reflexive Relations}=(\text{Choices for Diagonal Pairs})\times (\text{Choices for Non-Diagonal Pairs})$$

$$\text{Total Reflexive Relations}=(1{)}^n\times (2{)}^{n^2-n}=2^{n^2-n}$$

Step 4: Substitute $n=4$ into the Formula

Now, substitute the value of $n=4$ into our derived formula:

$$\text{Total Reflexive Relations}=2^{4^2-4}$$

$$\text{Total Reflexive Relations}=2^{16-4}$$

$$\text{Total Reflexive Relations}=2^{12}$$

Let's compute the value of $2^{12}$:

$$2^{10}=1024$$

$$2^{11}=2048$$

$$2^{12}=4096$$

Conclusion

The total number of possible reflexive relations on the set $A$ is $4096$.

Therefore, the correct option is (d).