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Unformatted text preview: tly stated. Easier to understand and to explain to people. But it is also easier to introduce errors. ༉ Proofs have many practical applications: ༉ Veriﬁcation that computer programs are correct. ༉ Enabling programs to make inferences in artiﬁcial intelligence. ༉ Showing that system speciﬁcations are consistent. Deﬁni$ons ༉ A theorem is a statement that can be shown to be true using: ༉ deﬁnitions, ༉ other theorems, ༉ axioms (statements which are given as true), and ༉ rules of inference. ༉ A lemma is a “helping theorem” or a result which is needed to prove a theorem. ༉ A corollary is a result which follows directly from a theorem. ༉ Less important theorems are sometimes called propositions. Forms of Theorems ༉ Many theorems assert that a property holds for all elements in a domain, such as the integers, the real numbers, or some of the discrete structures that we will study in this course. ༉ Often the universal quantiﬁer (needed for a precise statement of a theorem) is omitted by standard mathematical convention. For example, the statement: “If x > y, where x and y are positive real numbers, then x2 > y2 ” Really means “For all positive real numbers x and y, if x > y, then x2 > y2 .” Proving Theorems ༉ Many theorems have the form: ༉ To prove them, we show that where c is an arbitrary element of the domain, ༉ By univ...
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This document was uploaded on 02/27/2014 for the course CS 215 at SIU Carbondale.
 Spring '14
 M.Nojoumian
 Computer Science

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