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.