Let X be the set of all citizens of India. Elements x, y in X are said to be related if the difference of their age is 5 years. Which one of the following is correct?

Concept:

1. Reflexive: Each element is related to itself.

R is reflexive if for all x ∈ A, xRx.

2. Symmetric: If any one element is related to any other element, then the second element is related to the first.

R is symmetric if for all x, y ∈ A, if xRy, then yRx.

3. Transitive: If any one element is related to a second and that second element is related to a third, then the first element is related to the third.

R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.

4. R is an equivalence relation if A is nonempty and R is reflexive, symmetric and transitive.

Calculation:

Here, xRy → x - y = 5 years Where x,y ∈  X (citizen of India)

For Reflexive:

xRx → x - x ≠ 5, So its not reflexive

For symmetric:

xRy → x - y = 5 and yRx → y - x = 5, difference will be 5 years only

So its symmetric

For transitive:

Let, x = 25, y = 20, and z = 15

x - y = 5, y - z = 5, but x - z ≠ 5

If xRy and yRz, then not necessarily, xRz.

So, its not transitive.

Hence, option (2) is correct.