## Functions

 Question 1
How many onto (or surjective) functions are there from an n-element (n ≥ 2) set to a 2-element set?
 A 2n B 2n-1 C 2n-2 D 2(2n– 2)
Engineering-Mathematics       Functions       Gate 2012
Question 1 Explanation:
The number of onto functions from set of m elements to set of n elements, if m>n is
nm – (2n – 2)
i.e., 2n – (22 – 2) = 2n – 2
If there are 'm' elements in set A, 'n' elements in set B then
The number of functions are : nm
The number of injective or one-one functions are nPm
The number of surjective functions are:
If m If m>n, then n! * mCn
Given that m=n, n=2
2! * nC2
 Question 2

 A Theory Explanation is given below.
Engineering-Mathematics       Functions       Gate-2002
 Question 3
On the set N of non-negative integers, the binary operation __________ is associative and non-commutative.
 A fog
Engineering-Mathematics       Functions       Gate-1994
Question 3 Explanation:
The most important associative operation thats not commutative is function composition. If you have two functions f and g, their composition, usually denoted fog, is defined by
(fog)(x) = f(g(x))
It is associative, (fog)oh = fo(goh), but its usually not commutative. fog is usually not equal to gof.
Note that if fog exists then gof might not even exists.
 Question 4
The function f(x,y) = x2y - 3xy + 2y + x has
 A no local extremum B one local minimum but no local maximum C one local maximum but no local minimum D one local minimum and one local maximum
Engineering-Mathematics       Functions       Gate-1993
Question 4 Explanation:
Note: Out of syllabus.
 Question 5

 A Out of syllabus.
Engineering-Mathematics       Functions       Gate-1993
 Question 6
Let A and B be sets with cardinalities m and n respectively. The number of one-one mappings (injections) from A to B, when m < n, is:
 A mn B nPm C mCn D nCm
Engineering-Mathematics       Functions       Gate-1993
Question 6 Explanation:
Let,

A one-one function 'f' assigns each element ai of A a distinct element, bj=f(ai) of Bi for a, there are n choices, for a2 there are n-1 choices, for am there are (n-(m-1)) choices.
i.e.,
There are 6 questions to complete.