Match the following in List – I and List – II, for a function f:

List-I	List-II
(a) ∀x∀y (f(x) = f(y) → x = y)	(i) Constant
(b) ∀y∃x (f(x) = y)	(ii) Invective
(c) ∀x f(x) = k	(iii) Subjective

(A) a – (i), b – (ii), c – (iii)
(B) a – (iii), b – (ii), c – (i)
(C) a – (ii), b – (i), c – (iii)
(D) a – (ii), b – (iii), c – (i)


Answer: (D)

Explanation:

• ∀x∀y (f(x) = f(y) → x = y), that means if two functions maps same value then input of the functions should be same. This is definition of injective (or one-to-one) function.
  An injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.




• ∀y∃x (f(x) = y), that means for all y, there is a mapping function from x. This is definition of surjective (or onto) function.
  A function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y.




• ∀x f(x) = k, that means for all x, the output or mapping is only k and never changes. This is definition of constant function.
  A constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value.

Therefore, option (D) a – (ii), b – (iii), c – (i) is correct.


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