Proving condional statements p q reminder the real

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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.

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