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Adopting a trouble-free, conversational—and from time to time humorous—style, those authors make the rules and practices of discrete arithmetic as stimulating as attainable whereas featuring accomplished, rigorous assurance. Examples and workouts built-in all through each one bankruptcy serve to pique reader curiosity and produce readability to even the main advanced innovations. peculiarly, the booklet is designed to interact today's readers within the fascinating, acceptable features of contemporary arithmetic. greater than 2 hundred labored examples and difficulties, in addition to over 2500 workouts are integrated. complete recommendations are supplied at the back of the ebook. greater than a hundred and fifty *Pauses*—short questions inserted at strategic points—are incorporated. complete options to *Pauses* are incorporated on the finish of every part. For educators in zone of discrete arithmetic.

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**Additional info for Discrete Mathematics with Graph Theory (2nd Edition)**

**Example text**

Some Basic Logical Equivalences 1. Idempotence: (i) (pVp) p (ii) (p A p) p 2. Commutativity: (i) (p v q) (q v p) (ii) (p A q) (q A p) 3. '(p (ii) v (q V r)) (pA(q Ar)) ((pAq)A r) 4. Distributivity: (i) (p V (q A r)) . ((p v q) A (p V r)) (ii) (p A (q v r)) is((p A q) v (p A r)) 5. Double Negation: - (-p) 6. 1. It is clear that any two tautologies are logically equivalent and that any two contradictions are logically equivalent. Letting 1 denote a tautology and 0 a contradiction, we can add the following properties to our list.

From the Latin quod erat demonstrandum (which was to be proved) for the same purpose. In the index (see equivalent), you are directed to other places in this book where we establish the equivalence of a series of statements. " Sometimes, it is possible to prove such a statement directly; that is, by establishing the validity of a sequence of implications: A PROBLEM 2. AI -* -+ B. A2 Prove that for all real numbers x, x 2 - 4x + 17 #&0. 2 Solution. We observe that x - 4x + 17 = (x - 2)2 + 13 is the sum of 13 and a number, (x - 2)2, which is never negative.

The set of natural numbers, which is denoted with a capital boldface N or, in handwriting, with this symbol, N, consists of the numbers 1, 2, 3,... ' The elements of N are, of course, just the positive integers. The full set of integers, denoted Z or A, consists of the natural numbers, their negatives, and 0. We might describe this set by ... , -3, -2, -1, 0, 1, 2, 3 .... Our convention, which is not universal, is that 0 is an integer, but not a natural number. There are various ways to describe sets.