Logic

A Logical Introduction to Proof by Daniel W. Cunningham

By Daniel W. Cunningham

The publication is meant for college kids who are looking to find out how to turn out theorems and be larger ready for the pains required in additional improve arithmetic. one of many key elements during this textbook is the advance of a strategy to put naked the constitution underpinning the development of an evidence, a lot as diagramming a sentence lays naked its grammatical constitution. Diagramming an explanation is a fashion of proposing the relationships among some of the elements of an evidence. an explanation diagram presents a device for exhibiting scholars find out how to write right mathematical proofs.

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Then 1. {z ∈ U : z2 = 9} = {−3, 3}. 2. {z ∈ U : z2 ≤ 16} = {−3, −4, 0, −1, −2, 1, 2, 4, 3}. 3. {x ∈ U : x2 > 225} = {16, −16, −20, −19, −18, −17, 19, 20, 18, 17}. 3 Important Sets in Mathematics The reader should be familiar with the natural numbers, the integers, and the real numbers. In elementary algebra, we learned many properties about the real numbers, including the commutative and associative laws for multiplication and addition. You may have also learned that when a real number can be expressed as the ratio of two integers, then it is called a rational number.

Using quantifier negation laws and propositional logic laws, express each statement in as a positive statement. ) (a) ¬(∀x > 3)(|x − 10| < 12 → x2 − 100 < 13 ). 1 (b) ¬(∃x < −4)(|x + 6| < 100 ∧ |sin(x) − 100| ≥ 1 30 ). 46 2 Predicate Logic 5. Consider the Tarskian predicates in Example 5 on page 38. Using quantifier negation laws and propositional logic laws, express the negation of each of the following assertions as a positive statement. Then write your result in idiomatic English. (a) (b) (c) (d) (e) ∀x(I(x) → (T (x) ∨ S(x))).

4. 5. 6. Every cat is an animal. Some cat is an animal. No cat is an animal. Someone in this class does not do their homework. Everyone in this class does their homework. Nobody in this class does their homework. 2 Quantifiers 37 Solution. We will express the given six sentences into logical form. We first identify the two predicates that appear in sentences 1–3. ” 1. ” In logical form, we have ∀x(C(x) → A(x)). 2. ” In logical form, we have ∃x(C(x) ∧ A(x)). 3. There are two equivalent ways to restate sentence 3.

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