MATH 1190 Lecture Notes - Lecture 14: Universal Instantiation, Universal Generalization, Discrete Mathematics
12 Feb 2018
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Math 1190 lecture 14 notes - rules of inference for quantified statements, combining rules of inference for propositions and quantified statements, and introduction to proofs. Introduction: we will now describe some important rules of inference for statements involving quantifiers, these rules of inference are used extensively in mathematical arguments, often without being explicitly mentioned. D(marla) c(marla) ---------universal instantiation from (1: 3. These steps can be used to establish the conclusion from the premises: step ------- reason, 1. C(a) b(a)-------- existential instantiation from (1: 3. C(a) p (a) --------universal instantiation from (4) P (a) --------modus ponens from (3) and (5: 7. P (a) b(a)-------------- conjunction from (6) and (7: 9. X(p (x) b(x))--------- existential generalization from (8) To see this, note that by universal instantiation, p (a) q(a) is true. Let p (n) denote n > 4 and q(n) denote n2 < 2n. However, formal proofs of useful theorems can be extremely long and hard to follow.
