The University of Sydney
MATH 1004
Summer School
Discrete Mathematics
2015
Tutorial 9
1.
Draw truth tables for the following propositions.
(i ) (p q).
(ii ) p q.
(iii ) p (p q).
Solution.
The truth tables for the given propositions are shown as follows:
(
The University of Sydney
MATH 1004
Summer School
Discrete Mathematics
2015
Tutorial 8
1.
Find simple switching circuits corresponding to the following Boolean functions:
x
y
z
f (x, y, z)
1
1
1
1
0
0
0
0
(i)
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
(ii)
0
1
1
1
1
The University of Sydney
MATH 1004
Summer School
Discrete Mathematics
2015
Tutorial 5
1.
You have a deck of fty-two cards.
(i )
How many ways are there of choosing a hand of ve cards?
(ii ) How many of them contain the queen of hearts?
(iii ) In how many
The University of Sydney
MATH 1004
Summer School
Discrete Mathematics
2015
Tutorial 4
1.
(i )
If A = cfw_1, 2 and B = cfw_a, b, c, write down the set A B.
(ii ) For A = cfw_1, 2, 3, 4, write down the subset of A A consisting of all those
ordered pairs (x,
The University of Sydney
MATH 1004
Summer School
Discrete Mathematics
2015
Tutorial 2
1.
Use the notation of set theory to describe:
(i )
The set of all odd integers between 2 and 10.
(ii ) The set of all odd integers between 2 and 200.
(iii ) The set of
The University of Sydney
MATH 1004
Summer School
Discrete Mathematics
2015
Tutorial 6
1.
If possible, compute |AB C| from the given information. If it is not possible,
explain why.
(i )
|A| = 12, |B| = 13, |C| = 7, |A B| = 5, |A C| = 4, |B C| = 3 and
|A B
The University of Sydney
MATH 1004
Summer School
Discrete Mathematics
2015
Tutorial 3
1.
Dene f : N N by f (x) = x + 1. Determine whether or not f is
(a)
one-to-one;
(b)
onto.
Solution.
(a)
(b)
2.
If x = y, then clearly x + 1 = y + 1 so that f (x) = f (y)
The University of Sydney
MATH 1004
Second Semester
Discrete Mathematics
2015
Tutorial 12
1.
(i )
Show that xn = 6 2n 4 is a solution to the recurrence relation
xn = 3xn1 2xn2 .
(ii ) Show that xn = (3n+1 1)/2 is a solution to the recurrence relation
xn+1
The University of Sydney
MATH 1004
Summer School
Discrete Mathematics
2015
Tutorial 11
1.
Write a Boolean expression and the Boolean function for the switching circuit:
y
z
y
y
x
z
Solution.
The Boolean expression corresponding to the given switching circ
The University of Sydney
MATH 1004
Summer School
Discrete Mathematics
2015
Tutorial 10
Use induction to prove the following propositions.
For each of the given propositions, we let S(n) be the given proposition, which is
to be proved true for all integers
MATH1004
Summer School, 2015
Discrete Mathematics
Solutions to Tutorial 1
1. The rst few rows of Pascals triangle are written down below. Recall that each entry is
equal to the sum of the two numbers immediately above it.
n = 0:
1
n = 1:
1
n = 2:
1
n = 3: