Discrete Mathematics Multiple Choice Questions on “Functions”.
1. A function is said to be ______________ if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f.
a) One-to-many
b) One-to-one
c) Many-to-many
d) Many-to-one
Answer: b
Clarification: A function is one-to-one if and only if f(a)≠f(b) whenever a≠b.
2. The function f(x)=x+1 from the set of integers to itself is onto. Is it True or False?
a) True
b) False
Answer: a
Clarification: For every integer “y” there is an integer “x ” such that f(x) = y.
3. The value of ⌊1/2.⌊5/2⌋ ⌋ is ______________
a) 1
b) 2
c) 3
d) 0.5
Answer: a
Clarification: The value of ⌊5/2⌋ is 2 so, the value of ⌊1/2.2⌋ is 1.
4. Which of the following function f: Z X Z → Z is not onto?
a) f(a, b) = a + b
b) f(a, b) = a
c) f(a, b) = |b|
d) f(a, b) = a – b
Answer: c
Clarification: The function is not onto as f(a)≠b.
5. The domain of the function that assign to each pair of integers the maximum of these two integers is ___________
a) N
b) Z
c) Z +
d) Z+ X Z+
Answer: d
Clarification: The domain of the integers is Z+ X Z+.
6. Let f and g be the function from the set of integers to itself, defined by f(x) = 2x + 1 and g(x) = 3x + 4. Then the composition of f and g is ____________
a) 6x + 9
b) 6x + 7
c) 6x + 6
d) 6x + 8
Answer: a
Clarification: The composition of f and g is given by f(g(x)) which is equal to 2(3x + 4) + 1.
7. __________ bytes are required to encode 2000 bits of data.
a) 1
b) 2
c) 3
d) 8
Answer: b
Clarification: Two bytes are required to encode 2000 (actually with 2 bytes you can encode up to and including 65,535.
8. The inverse of function f(x) = x3 + 2 is ____________
a) f -1 (y) = (y – 2) 1/2
b) f -1 (y) = (y – 2) 1/3
c) f -1 (y) = (y) 1/3
d) f -1 (y) = (y – 2)
Answer: b
Clarification: To find the inverse of the function equate f(x) then find the value of x in terms of y such that f -1 (y) = x.
9. The function f(x) = x3 is bijection from R to R. Is it True or False?
a) True
b) False
Answer: a
Clarification: The function f(x) = x3 is one to one as no two values in domain are assigned the same value of the function and it is onto as all R of the co domain is images of elements in the domain.
10. The g -1({0}) for the function g(x)= ⌊x⌋ is ___________
a) {x | 0 ≤ x < 1}
b) {x | 0 < x ≤ 1}
c) {x | 0 < x < 1}
d) {x | 0 ≤ x ≤ 1}
Answer: d
Clarification: g({0}) for the function g(x) is {x | 0 ≤ x ≤ 1}. Put g(x) = y and find the value of x in terms of y such that ⌊x⌋ = y.