Discrete Mathematics Multiple Choice Questions on “Number of Functions”.
1. An injection is a function which is?
a) many-one
b) one-one
c) onto
d) none of the mentioned
Answer: b
Clarification: One-One functions are also known as injection.
2. A mapping f : X → Y is one one if __________
a) f(x1) ≠ f(x2) for all x1, x2 in X
b) If f(x1) = f(x2) then x1 = x2 for all x1, x2 in X
c) f(x1) = f(x2) for all x1, x2 in X
d) None of the mentioned
Answer: b
Clarification: In one one function every element in A should have unique image in B,thus if two image are equal this means there preimages are same.
3. A function is defined by mapping f : A → B such that A contains m elements and B contains n elements and m ≤ n then number of one one functions are _________
a) nCm x m!
b) nCm x n!
c) 0
d) none of the mentioned
Answer: a
Clarification: From n elements in B we need to select m elements and then arrange them in all ways, thus answer=nCm x m!.
4. A function is defined by mapping f : A -> B such that A contains m elements and B contains n elements and m>n then number of one one functions are ________
a) nCm x m!
b) nCm x n!
c) 0
d) none of the mentioned
Answer: c
Clarification: Since for function to be defined every element should have a image, since m > n atleast one element need to have same image, thus we can’t form any unique images and hence the number of one one function are zero.
5. For an onto function range is equivalent to codomain.
a) True
b) False
Answer: a
Clarification: Since in onto function every image should have preimage thus all the elements in codomain should have preimages.
6. Onto function are known as injection.
a) True
b) False
Answer: b
Clarification: Onto functions are known as surjection.
7. Set A has 3 elements and set B has 4 elements then number of injections defined from A to B are?
a) 12
b) 24
c) 36
d) 48
Answer: b
Clarification:Injections will be 4C3 x 3!=24.
8. A function is defined by mapping f : A → B such that A contains m elements and B contains n elements and 1≤n≤m then number of onto functions are ________
a) r=1∑r=nnCr (-1)n-r rm
b) r=1∑r=nnCr (-1)n-r rn
c) r=1∑r=nnCr (-1)m-r rn
d) None of the mentioned
Answer: a
Clarification: The number of onto function is equal tpo the coffecient of xm in m!(ex – 1)n.
9. A function is defined by mapping f:A→B such that A contains m elements and B contains n elements and m > n then number of bijections are ________
a) nCm x m!
b) nCm x n!
c) 0
d) none of the mentioned
Answer: c
Clarification: Since we can’t define any one one function in such case so number of bujections will be zero.
10. A bijection is a function which is many-one and onto.
a) True
b) False
Answer: b
Clarification: A bijection is a function which is one-one(injection) and onto(surjection).