# 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 &gt; y, where x and y are positive real numbers, then x2 &gt; y2 ” Really means “For all positive real numbers x and y, if x &gt; y, then x2 &gt; 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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