Consider the following statements :

1. If f is the subset of Z × Z defined by f = {(xy, x − y); x, y ∈ Z}, then f is a function from Z to Z.

2. If f is the subset of N × N defined by f = {(xy, x + y); x, y ∈ N}, then f is a function from N to N.

Which of the statements given above is/are correct?

Concept:

A relation f from set A to set B is said to be a function if every element of set A has a unique image in set B.

Calculation:

1. If f is the subset of Z × Z defined by f = {(xy, x − y); x, y ∈ Z},

let x = 2, y = 3 → (2 × 3, 2 - 3) = (6, -1) ∈ f

and for x = -2, y = -3 → (-2 × (-3), -2 + 3) = (6, 1) ∈ f

So, the element 6 ∈ Z has two different images 1 and -1 in Z 

⇒ f is not a function.

2. If f is the subset of N × N defined by f = {(xy, x + y); x, y ∈ N},

 let x = 3, y = 4→ (3 × 4, 3 + 4) = (12, 7) ∈ f

and for x = 2, y = 6 → (2 × (6), 2 + 6) = (12, 8) ∈ f

So, the element 12 ∈ N has two different images 7 and 8 in N

⇒ f is not a function.

∴ The correct option is (4).