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Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians.
We have seen that a statement P(x) containing a free variable x may be true for some values of x and false for others. Sometimes we want to say something about how many values of x make P(x) come out true. In particular, we often want to say either that P(x) is true for every value of x or that it is true for at least one value of x. We therefore introduce two more symbols, called quantifiers, to help us express these ideas.
To say that P(x) is true for every value of x in the universe of discourse U, we will write ∀x P(x). This is read “For all x, P(x).” Think of the upside down A as standing for the word all. The symbol ∀ is called the universal quantifier, because the statement ∀x P(x) says that P(x) is universally true. As we discussed in Section 1.3, to say that P(x) is true for every value of x in the universe means that the truth set of P(x) will be the whole universe U. Thus, you could also think of the statement ∀x P(x) as saying that the truth set of P(x) is equal to U.
We write ∃x P(x) to say that there is at least one value of x in the universe for which P(x) is true.