By Patrick Murphy
ISBN-10: 0434985457
ISBN-13: 9780434985456
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Dieses erfolgreiche einf? hrende Lehrbuch liegt nun in der 10. Auflage vor. Es zeichnet sich durch eine exakte und anschauliche Darstellung aus. Der Lehrstoff ist klar gegliedert und intestine strukturiert. Er wird durch eine F? lle von Beispielen und Abbildungen veranschaulicht und vertieft. Zahlreiche Aufgaben mit L?
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Two integers m and η do not have a common factor. Prove that it is not possible to have 2m^ = n^. ) Mathematical Sentences We describe ideas with the aid of sentences. In everyday events and language we use sentences like ΊΧ is raining', T h e bus is late' and sometimes we might say these sentences whether they are true or false. ' are neither true nor false. In mathematics we use sentences like 'Five is the greatest number in the set', 'Thirty is four more than twenty' and, as with the previous examples, these may be true or false.
If sets A, Β are disjoint (T) then AnB = AuB. 8. If X # 0 and X < 0 then χ > 0. 9. For a quadrilateral to be a square it is sufficient that all of the angles be right-angles. 10. A necessary condition for a quadrilateral to be a square is that all its sides be of equal length. Bicoiiditioiuil Sentences As may be expected by the name, a biconditional χ y is an implication which is read two ways—^that is, not only does χ imply y but y implies x. The notation of χ y for a biconditional is therefore very appropriate.
Thus, even though we had a million examples to support the claim this was still not enough to prove the claim. To illustrate the point of being able to exhaust all possibilities consider the following example. Example 1. Prove that the number 359 873 cannot be a perfect square. Solution. Any positive integer must end in one of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (i) When such a number is squared, the result will end in one of the digits 0,1,4,9,6,5,6,9,4,1 (ii) Hence, no squared number ends in 3 and so 359 873 cannot be a perfect square.