CmSc 180 – Discrete mathematics
Problems on Functions
1.
Give an example of a function that is one-to-one but not onto
2.
Give an example of a function that is onto but not one-to-one.
3.
Give an example of a function that is neither one-to-one nor onto
4.
Give an example of a function that is both one-to-one and onto
5.
How many functions are there from A = {1,2} to B = {a, b}? Write them as sets
of ordered pairs. Which are one-to-one? Which are onto?
6.
Let X = {1, 2, 3, 4}, Y = {a, b, c, d}. For each of the following subsets of
X x Y
determine whether it is a function or not. If it is a function, determine whether it is
one-to-one, onto, or both. If it is a bijection, determine its inverse function as a set
of ordered pairs.
A1 = {(1,a), (2,a), (3,c), (4, b)}
A2 = {(1, c), (2, a), (3, b), (4, c), (2, d)}
A3 = {(1, c), (2, d),
(3, a), (4, b)}
A4 = {(1, d), (2, d), (4, a)}
A5 = {(1, b), (2, b), (3, b), (4, b)}
7.
Do the following sets define functions? If so, give their domain and range:
F1 = {(1, (2,3)), (2, (3,4)), (3, (1,4)), (4, (2,4))}
F2 = {((1,2), 3), ((2,3), 4), ((3,3), 2)}
F3 = {(1, (2,3)), (2, (3,4)), (1, (2,4))}
F4 = {(1, (2,3)), (2, (2,3)), (3, (2,3))}
8.
Let N be the set of all non-negative integers. Determine which of the following
functions are one-to-one, which are onto, and which are one-to-one and onto:
a.