By Daniel W. Cunningham
The ebook is meant for college students who are looking to tips on how to end up theorems and be greater ready for the trials required in additional improve arithmetic. one of many key parts during this textbook is the improvement of a strategy 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 fashion of proposing the relationships among a few of the elements of an explanation. an explanation diagram offers a device for exhibiting scholars find out how to write right mathematical proofs.
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Extra info for A Logical Introduction to Proof
Thus, 32 ∈ Q. R is the set of real numbers and so, π ∈ R. The set N is closed under the operations of addition and multiplication, that is, the sum and product of two natural numbers is a natural number. Moreover, the sets Z, Q, and R are closed under addition, multiplication, and subtraction. For example, if we add, multiply, or subtract any two rational numbers the result is again a rational number. Finally, recall that each nonzero element in Q, and R, has a multiplicative inverse. For example, if x ∈ Q and x = 0, then there is a y ∈ Q such that x · y = 1.
LOGICAL NEGATION: ¬∀x(C(x) → A(x)). ¬∀x(C(x) → A(x)) ⇔ ∃x¬(C(x) → A(x)) by Quantifier Negation Law ⇔ ∃x(C(x) ∧ ¬A(x)) by Conditional Law. ” 2. ” LOGICAL FORM: ¬∃x(C(x) ∧ A(x)). ” LOGICAL NEGATION: ¬¬∃x(C(x) ∧ A(x)). ¬¬∃x(C(x) ∧ A(x)) ⇔ ∃x(C(x) ∧ A(x)) by Double Negation Law. ” 44 2 Predicate Logic 3. ” LOGICAL FORM: ∃x(C(x) ∧ ¬H(x)). ” LOGICAL NEGATION: ¬∃x(C(x) ∧ ¬H(x)). ¬∃x(C(x) ∧ ¬H(x)) ⇔ ∀x¬(C(x) ∧ ¬H(x)) by Quantifier Negation Law ⇔ ∀x(C(x) → ¬¬H(x)) by Conditional Law ⇔ ∀x(C(x) → H(x)) by Double Negation Law.
The words sufficient and necessary can also be confusing. One way to avoid this confusion is to think of the word “sufficient” as the arrow →, and to think of the word “necessary” as the backward arrow ←. Our next three logic laws involve conditional statements. The first law states that a conditional statement is equivalent to one that contains the connectives ¬ and ∨. Conditional Laws 1. (P → Q) ⇔ (¬P ∨ Q). 2. (P → Q) ⇔ ¬(P ∧ ¬Q). 3. ¬(P → Q) ⇔ (P ∧ ¬Q). Proof of Conditional Laws. We show that items 1 and 2 hold, by comparing the following truth tables P Q P→Q P Q ¬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 T F T T T T F F T F T F T F T T Since all of the final columns agree, we see that (P → Q), (¬P ∨ Q), and ¬(P ∧ ¬Q) are logically equivalent.
A Logical Introduction to Proof by Daniel W. Cunningham