Sets And Functions

Question 1
 
A
3m
B
3n
C
2m+1
D
2n+1
       Engineering-Mathematics       Sets And Functions       Gate-2006
Question 1 Explanation: 
No. of subsets of size 3 is = mC3
n = mC3
Which subsets contains element i then size is
= (m-1)C2
Because 1 element is already known
Question 2
Let f: B → C and g: A → B be two functions and let h = f∘g. Given that h is an onto function. Which one of the following is TRUE?
A
f and g should both be onto functions
B
f should be onto but g need not be onto
C
g should be onto but f need not be onto
D
both f and g need not be onto
       Engineering-Mathematics       Sets And Functions       Gate-2005
Question 2 Explanation: 
Given:
f: B→C and g: A→B are two functions.
h = f∘g = f(g(x))
→ If his onto function, that means for every value in C, there must be value in A.
→ Here, we are mapping C to A using B, that means for every value in C there is a value in B then f is onto function.
→ But g may (or) may not be the onto function i.e., so values in B which may doesn't map with A.
Question 3
 
A
g(h(D)) ⊆ D
B
g(h(D)) ⊇ D
C
g(h(D)) ∩ D = ɸ
D
g(h(D)) ∩ (B—D) ≠ ɸ
       Engineering-Mathematics       Sets and Functions       Gate-2003
Question 3 Explanation: 
f: A→B ba an injective (one-to-one)
→ g: 2A→2B be also one to one function and g(C) = f(x)|x∈C}, for all subsets C of A.
The range of this function is n(2A).
→ h: 2B→2A it is not a one to one function and given h(D) = {x|x∈A, f(x)∈D}, for all subsets D of B.
The range of this function is also n(2A).
→ The function g(h(D)) also have the range n(2A) that implies n(A)≤n(B), i.e., n(2A) is less than n(2B).
Then this result is g(h(D))⊆D.
Question 4
The number of functions from an m element set to an n element set is
A
m + n
B
mn
C
nm
D
m*n
       Engineering-Mathematics       Sets and Functions       Gate-1998
Question 4 Explanation: 
Here each m element we have n choices i.e.,
n×n×n×n×...×n (m times)
= nm
Question 5
Suppose X and Y are sets and X Y and are their respective cardinalities. It is given that there are exactly 97 functions from X to Y. from this one can conclude that  
A
|X| = 1, |Y| = 97
B
|X| = 97, |Y| = 1
C
|X| = 97, |Y| = 97
D
None of the above
       Engineering-Mathematics       Sets and Functions       Gate-1996
Question 5 Explanation: 
From the given information we can write,
|Y||X| = 97
→ Option A only satisfies.
Question 6
Let R denotes the set of real numbers. Let f: R×R → R×R be a bijective function defined by f(x, y) = (x+y, x-y). the inverse function of f is given by  
A
B
C
D
       Engineering-Mathematics       Sets and Functions       Gate-1996
Question 6 Explanation: 
Question 7
 
A
Theory Explanation is given below.
       Engineering-Mathematics       Sets and Functions       Gate-2001
There are 7 questions to complete.