# UGC-NET | UGC NET CS 2018 July – II | Question 87

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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