JEE Mathematics 2023 Question with Solution

Given: $A = \{-4, -3, 2, 0, 1, 3, 4\}$ and $R = \{(a,b): a \in A, b = |a| \text{ or } a=b\}$.

Find: The minimum number of ordered pairs to be added so that $R$ becomes reflexive and symmetric.

From the given relation, the pairs generated are

$$R = \{(-4,4), (-4,-4), (-3,3), (2,2), (0,0), (1,1), (3,3), (4,4)\}$$

For a relation to be reflexive, every element of $A$ must satisfy $(a,a) \in R$. Thus we need

$$(-3,-3)$$

to be added, since all other diagonal pairs are already present.

For a relation to be symmetric, whenever $(a,b) \in R$, we must also have $(b,a) \in R$. The non-symmetric pairs are:

• $(-4,4)$, whose reverse $(4,-4)$ is missing
• $(-3,3)$, whose reverse $(3,-3)$ is missing

So we must add

$$(4,-4), (3,-3), (-3,-3)$$

But $(-3,-3)$ has already been counted for reflexivity, so the distinct pairs to add are

$$(-3,-3), (4,-4), (3,-3)$$

Hence the minimum number of elements to be added is $3$.

The solution concludes $7$, but its listed relation and reasoning are internally inconsistent. Based on the given definition of $R$, the correct numerical answer is $3$.