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Unformatted text preview: set S to be the
smallest set contained in N and satisfying the following two properties:
S, and
S, then n ¢ 2. whenever n 3 ¢ ¢ 1. 0 S. Deﬁnition 3 (Example of deﬁnition by rules of inference) S 3) ¢ (n S ¢ n ¢ 0 S Each entry is called a rule of inference, or just a rule; the horizontal line is
read as an “ifthen”. The part above the line is called the hypothesis or the antecedent; the part
below the line is called the conclusion or the consequent. When there are two or more hypotheses listed, they are connected by an
implicit “and.” A rule with no hypotheses is called an axiom.
The rules are interpreted as saying that an integer n is in S if and only if
the statement “n S” can be derived from the axioms by using the rules of
inference ﬁnitely many times. ¢ These three deﬁnitions are entirely equivalent. We’ll move back and forth between
them freely. 6 Deﬁnition 4 (list of integers, topdown) A Scheme list is a list of integers if and
only if either
1. it is the empty lis...
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This document was uploaded on 03/17/2014 for the course CSG 111 at Northeastern.
 Fall '09

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