Logic

Get A Logical Introduction to Proof PDF

By Daniel W. Cunningham

ISBN-10: 1461436303

ISBN-13: 9781461436300

ISBN-10: 1461436311

ISBN-13: 9781461436317

The ebook is meant for college students who are looking to how you can turn out theorems and be greater ready for the pains required in additional develop arithmetic. one of many key elements during this textbook is the advance of a technique to put naked the constitution underpinning the development of an explanation, a lot as diagramming a sentence lays naked its grammatical constitution. Diagramming an explanation is a manner of proposing the relationships among many of the elements of an evidence. an evidence diagram offers a device for exhibiting scholars how one can write right mathematical proofs.

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Additional info for A Logical Introduction to Proof

Example text

P(x) ∧ D(x, y). If x = 4 and y = 8, then the statement is true. When x = 3 and y is any integer, the statement is false. 4. D(x, y) → ¬P(x). When x = 5 and y = 10, the statement is true. If x = 1 and y = 3, then the statement is false. Example 2. Analyze the logical forms of the following statements, that is, write each statement symbolically, using the predicates P, E, D defined in Example 1. 1. x is a prime number, and either y is even or z is divisible by x. 2. Exactly one of x and y is even.

Biconditional Law 1. (P ↔ Q) ⇔ (P → Q) ∧ (Q → P). 5, to derive new logic laws. Example 2. Show that (P → R) ∧ (Q → R) ⇔ (P ∨ Q) → R, by using propositional logic laws. Solution. We first start with the more complicated side (P → R) ∧ (Q → R) and derive the simpler side as follows: (P → R) ∧ (Q → R) ⇔ (¬P ∨ R) ∧ (¬Q ∨ R) by Conditional Law(1) ⇔ (¬P ∧ ¬Q) ∨ R by Distributive Law(4) ⇔ ¬(P ∨ Q) ∨ R by De Morgan’s Law(1) ⇔ (P ∨ Q) → R by Conditional Law(1). Therefore, (P → R) ∧ (Q → R) ⇔ (P ∨ Q) → R.

The logical form of statement 1 is P(x) ∧ (E(y) ∨ D(x, z)). ” We get E(x) ∨ E(y) for the “one or the other” part, and for the “not both” part we get ¬(E(x) ∧ E(y)). Thus, the logical form of the given statement is (E(x) ∨ E(y)) ∧ ¬(E(x) ∧ E(y)). 1 Universe of Discourse We say that A is a set when A is a collection of objects. The objects that belong to a set A are called the elements of A. We write a ∈ A to mean that a is an element, or a member, of the set A. We write a ∈ / A when a is not an element of the set A.

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A Logical Introduction to Proof by Daniel W. Cunningham


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