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[eBook] Discrete Mathematics - W W L CHEN

# [eBook] Discrete Mathematics - W W L CHEN - DISCRETE...

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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 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. 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 first used in lectures given by the author at Imperial College, University of London, in 1982.

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1–2 W W L Chen : Discrete Mathematics 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 = 4”. Example 1.1.6. The negation of the sentence “ π is rational” is the sentence “ π is irrational”. Definition. (CONDITIONAL) We say that the sentence p q (if p , then q ) is true if the sentence p is false or if the sentence q is true or both, and is false otherwise.
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