A Logical Introduction to Proof by Daniel W. Cunningham PDF

By Daniel W. Cunningham

ISBN-10: 1461436303

ISBN-13: 9781461436300

The booklet is meant for college students who are looking to turn out theorems and be greater ready for the pains required in additional boost arithmetic. one of many key elements during this textbook is the improvement 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 evidence is a fashion of featuring the relationships among a few of the elements of an explanation. an evidence diagram presents a device for displaying scholars the way to write right mathematical proofs.

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Negation laws, similar to those for bounded set quantifiers, also apply to the bounded number quantifiers. For example, let R be our universe and let P(x) be a predicate. Note that the statement ¬(∀x > a)P(x) asserts that not every real number x > a satisfies P(x). This just means (∃x > a)¬P(x); that is, there is a real number x > a that fails to satisfy P(x). Therefore, ¬(∀x > a)P(x) ⇔ (∃x > a)¬P(x). 3 Quantifiers and Negation 45 Negation Laws for Bounded Number Quantifiers 1. 2. 3. 4. ¬(∀x > a)P(x) ⇔ (∃x > a)¬P(x).

Similarly, the superscript − means that only the negative numbers are permitted. For example, 1. Q+ = {x ∈ Q : x > 0}. 2. Z− = {x ∈ Z : x < 0}. 3. R+ = {x ∈ R : x > 0}. For sets A and B we write A ⊆ B to mean that the set A is a subset of the set B, that is, every element of A is also an element of B. Thus, N ⊆ Z. Example 4. Consider the following three subsets of Z: 1. {x ∈ Z : x is a prime number} = {2, 3, 5, 7, 11, . }. 2. {x ∈ Z : x is divisible by 3} = {. . , −12, −9, −6, −3, 0, 3, 6, 9, 12, .

This element y is usually denoted by 1x . 32 2 Predicate Logic For each of the sets Z, Q and R, we may add ‘+’ or ‘−’ as a superscript. The superscript + indicates that only the positive numbers will be allowed. Similarly, the superscript − means that only the negative numbers are permitted. For example, 1. Q+ = {x ∈ Q : x > 0}. 2. Z− = {x ∈ Z : x < 0}. 3. R+ = {x ∈ R : x > 0}. For sets A and B we write A ⊆ B to mean that the set A is a subset of the set B, that is, every element of A is also an element of B.

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