A Logical Introduction to Proof by Daniel W. Cunningham

A Logical Introduction to Proof by Daniel W. Cunningham

By Daniel W. Cunningham

The ebook is meant for college kids who are looking to how you can turn out theorems and be greater ready for the pains required in additional strengthen 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 evidence, a lot as diagramming a sentence lays naked its grammatical constitution. Diagramming an explanation is a manner of offering the relationships among a number of the components of an evidence. an explanation diagram presents a device for displaying scholars how you can write right mathematical proofs.

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1 (c) (∃x ∈ Z)( 5+x ∈ N). 1 (d) (∃x ∈ N)( 5+x ∈ Z). 42 2 Predicate Logic 9. Evaluate the truth sets: (a) (b) (c) (d) (e) (f) (g) (h) {x ∈ R : (∃y ∈ R)(x = y2 )}. {x ∈ R : (∀y ∈ R)(x < y2 )}. {x ∈ R : (∀y > 2)(x < y2 + 1)}. {x ∈ R : (∃y > 2)(x < y2 + 1)}. {x ∈ Z : (∃y ∈ Z)(x = y2 )}. {w ∈ Z : (∃x ∈ Z)(w = 3x)}. {q ∈ Q : (∃x ∈ Q+ )(qx = 1)}. {q ∈ Q : (∀x ∈ Q)(qx = x)}. 3 Quantifiers and Negation In this section we introduce laws that involve the negation of a quantified assertion. These laws are very useful when dealing with the denial of a complicated mathematical statement.

If an argument is valid and all of its premises are true, then we can be assured that the conclusion is also true. In other words, an argument is valid if it is impossible for the premises to be true and the conclusion to be false at the same time. In our next example, we present an inference rule that allows you to conclude that Q is true if you know that P and P → Q are true. Example 1. Show that the following argument is valid. P→Q P ∴Q Solution. The following truth table shows that the argument is valid Premise 1 Premise 2 Conclusion P Q (P → Q) P Q T T F F T F T F T F T T T T F F T F T F because whenever the premises are all true, the conclusion is also 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. Solution. The logical form of statement 1 is P(x) ∧ (E(y) ∨ D(x, z)).

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