CmSc 180 – Discrete mathematics
Homework 07 Solutions
1. Give an example of a function that is one-to-one but not onto
Let A = {1,2}, B = {a, b, c}
f: A  B f = {(1,a), (2,b)}
2. Give an example of a function that is onto but not one-to-one.
Let A = {1,2,3}, B = {a, b}
f: A  B f = {(1,a), (2,b), (3,b)}
3. Give an example of a function that is neither one-to-one nor onto
Let A = {1,2,3}, B = {a, b, c}
f: A  B f = {(1,a), (2,b), (3,b)}
4. Give an example of a function that is both one-to-one and onto
Let A = {1,2,3}, B = {a, b, c}
f: A  B f = {(1,a), (2,b), (3,c)}
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?
Each function is a subset of the Cartesian product A x B. Therefore all functions
are elements of the power set of A x B
A x B = {(1,a), (1,b), (2, a), (2, b)}
P(A x B) =
{ ,
{(1,a)},
{(1,b)},
{(2, a)},
{(2, b)},
{(1,a), (1,b)},
{(1,a), (2, a)},
{(1,a), (2, b)},
{(1,b), (2, a)},
{(1,b), (2, b)},
{(2,a), (2, b)},
{(1,a), (1,b), (2, a)},
{(1,a), (1,b), (2, b)},
not a function
not a function
not a function
not a function
not a function
not a function
function
function, one-to-one, onto
function, one-to-one, onto
function
not a function
not a function
not a function
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{(1,a), (2,a), (2, b)},
not a function
{(1,b), (2,a), (2, b)},
not a function
{(1,a), (1,b), (2, a), (2, b)} } not a function
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)}
Record the solution in the table below; check the boxes “function”, “1 – 1”, and
“onto” if the set has the corresponding properties. Write the inverse in the last
column, if the function is a bijection (i.e. both 1 – 1 and onto)
A1 = {(1,a), (2,a),
function
X
1-1
onto
Inverse (if present)
X
X
X
A3-1 =
{ (a, 3), (b, 4),
(c, 1), (d, 2) }
(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),
X
(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))}
Function, Domain : {1, 2, 3, 4}, Range: {(2,3), (3,4), (1,4), (2,4)}
2
F2 = {((1,2), 3), ((1,3), 4), ((1,3), 2)} not a function
F3 = {(1, (2,3)), (2, (2,3)), (1, (2,4))} not a function
F4 = {(1, (3,3)), (2, (3,3)), (3, (2,3))}
Function, Domain : {1, 2, 3}, Range: { (2, 3), (3,3)}
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. f: N N
f(n) = n2 + 2
one-to-one, not onto
b. f: N N
f(n) = n(mod 5)
not one-to-one, not onto
c. f: N N
f(n) = 1 if n is odd, 0 if n is even , not one-to-one, not onto
d. f: N {0, 1} f(n) = 1 if n is odd, 0 if n is even, not one-to-one, onto
9. Let X and Y be finite sets. Find a necessary condition for the existence of one-toone mappings from X to Y.
The number of elements in Y must be greater or equal to the number of
elements in X
10. Show that there exists a one-to-one function from A x B to B x A. Is it also onto?
A x B = {(a, b) | a A  bB}
B x A = {(b, a) | a A  bB}
Define f: A x B B x A = {((a,b),(b,a)) | (a,b)  A x B , (b,a)  B x A}
It is one-to-one and onto.
One-to-one: Let (a,b)  (c,d)
then (b, a)  (d, c) (1)
By our definition of f: f((a,b)) = (b,a), f((c,d)) = (d,c)
from (1): f((a,b))  f((c,d))
onto: show that y  B x A  x  A x B such that f(x) = y
Let y = (y1, y2)  B x A. (1)
From (1) and by def of Cartesian product B x A: y1  B, y2 A (2)
From (2) and by def of Cartesian product A x B : (y2,y1)  A x B (3)
From (3) and by def. of f: f((y2,y1)) = (y1,y2).
The choice of y = (y1,y2) was arbitrary.
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Thus y = (y1,y2) B x A  x = (y2,y1)  A x B such that f(x) = y
Therefore any element in B x A is an image of some element in A x B.
The next problems refer to composition of functions. By definition
f  g = g(f(x))
11. Let g = {(1, a), (2, b), (3, a)} be a function from X = {1, 2, 3} to Y = {a, b, c, d},
and f = {(a, w), (b, x), (c, z), (d, y)} be a function from Y to Z = {w, x, y, z}.
Write g  f as a set of ordered pairs.
g  f = f(g(x)) = {(1,w), (2, x), (3,w)}
12. Let f and g be functions from N to N defined by the equations:
f(n) = 2n, g(n) = n +1
Find the compositions f  f , g  g, g  f , and f  g
f  f = f(f(n)) = 2(2n) = 4n
g  g = g(g(n)) = (n + 1) + 1 = n + 2
g  f = f (g(n)) = 2(n + 1) = 2n + 2
f  g = g(f(n)) = (2n) + 1 = 2n + 1
13. Let f and g be functions from N to N defined by the equations:
f(n) = n2 , g(n) = 2n
Find the compositions f  f , g  g, g  f , and f  g
f  f = f(f(n)) = (n2 ) 2 = n4
g  g = g(g(n)) = 2^( 2n ) (the symbol ^ denotes the operation “raising to power”
g  f = f (g(n)) = (2n )2 = 22n
f  g = g(f(n)) = 2^( n2 ) = 2 n*n
14. Represent each function below as a composition of the functions
g(x) = x + 2
h(x) = sin(x)
k(x) = 2x
m(x) = cos(x)
n(x) = log(x)
p(x) = x 3
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a. f(x) = log 2 (x + 2)
f(x) = log 2 (x + 2) = n  g = n(g(x))
b. f(x) = sin(2x) = h  k = h(k(x))
c. f(x) = (2 + sin (x) ) 3
= p  g  h = p(g(h(x))
d. f(x) = 2sin(x) = k  h = k(h(x))
e. f(x) = (cos(x))3
= p  m = p(m(x))
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