DISCRETE MATHEMATICS
W W L CHEN
c
°
W W L Chen, 1982.
This work is available free, in the hope that it will be useful.
Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including
photocopying, recording, or any information storage and retrieval system, with or without permission from the author.
Chapter 1
LOGIC AND SETS
1.1. Sentences
In this section, we look at sentences, their truth or falsity, and ways of combining or connecting sentences
to produce new sentences.
A sentence (or proposition) is an expression which is either true or false. The sentence “2 + 2 = 4”
is true, while the sentence “
π
is rational” is false. It is, however, not the task of logic to decide whether
any particular sentence is true or false. In fact, there are many sentences whose truth or falsity nobody
has yet managed to establish; for example, the famous Goldbach conjecture that “every even number
greater than 2 is a sum of two primes”.
There is a defect in our deFnition. It is sometimes very diﬃcult, under our deFnition, to determine
whether or not a given expression is a sentence. Consider, for example, the expression “I am telling a
lie”; am I?
Since there are expressions which are sentences under our deFnition, we proceed to discuss ways of
connecting sentences to form new sentences.
Let
p
and
q
denote sentences.
Definition.
(CONJUNCTION) We say that the sentence
p
∧
q
(
p
and
q
) is true if the two sentences
p
,
q
are both true, and is false otherwise.
Example 1.1.1.
The sentence “2 + 2 = 4 and 2 + 3 = 5” is true.
Example 1.1.2.
The sentence “2 + 2 = 4 and
π
is rational” is false.
Definition.
(DISJUNCTION) We say that the sentence
p
∨
q
(
p
or
q
) is true if at least one of two
sentences
p
,
q
is true, and is false otherwise.
Example 1.1.3.
The sentence “2 + 2 = 2 or 1 + 3 = 5” is false.
†
This chapter was Frst used in lectures given by the author at Imperial College, University of London, in 1982.