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DISCRETE MATHEMATICS
W W L CHEN
c
±
W W L Chen, 1982, 2008.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for ﬁnancial gain,
and may be downloaded and/or photocopied, with or without permission from the author.
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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 deﬁnition. It is sometimes very diﬃcult, under our deﬁnition, 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 deﬁnition, 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.
Chapter 1 : Logic and Sets
page 1 of 9
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View Full Document Discrete Mathematics
c
±
W W L Chen, 1982, 2008
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.
Example 1.1.4.
The sentence “2 + 2 = 4 or
π
is rational” is true.
Remark.
To prove that a sentence
p
∨
q
is true, we may assume that the sentence
p
is false and use this
to deduce that the sentence
q
is true in this case. For if the sentence
p
is true, our argument is already
complete, never mind the truth or falsity of the sentence
q
.
Definition.
(NEGATION) We say that the sentence
p
(not
p
) is true if the sentence
p
is false, and is
false if the sentence
p
is true.
Example 1.1.5.
The negation of the sentence “2 + 2 = 4” is the sentence “2 + 2
6
= 4”.
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This note was uploaded on 12/20/2009 for the course MATH 245 taught by Professor ramaswamy during the Spring '08 term at San Diego State.
 Spring '08
 RAMASWAMY
 Math

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