By H. F. Mattson
Applauded through reviewers for its inviting, conversational variety and extraordinary assurance of good judgment and inductions, it introduces scholars to the subjects and language of discrete arithmetic and prepares them for destiny paintings in arithmetic and/or machine technology. Mattson develops scholars' mathematical pondering and total adulthood via cautious presentation and improvement of proofs, various distinctive examples and corresponding routines and purposes that permit scholars to make concrete use of the idea awarded. routines are assorted, starting from basic difficulties to difficult extensions of the themes brought.
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Extra resources for Discrete Mathematics with Applications
The commutative law in logic is comparable to the commutative law of addition and multiplication in algebra; that is, a + b5b + a and a×b5b×a. 4 Distributive p ∨ ðq ∧ rÞ≡ðp∨ qÞ∧ ðp∨ rÞ and p ∧ ðq ∨ rÞ≡ðp∧ qÞ∨ ðp∧ rÞ Distributive describes the property of one operator to be expanded in a particular way which yield the same result; that is, an equivalent expression. The distributive law in logic is more extensive than the distributive property in algebra. ” However, the idea of distribution of one operator over a second operator is comparable; for multiplication over addition we have aðb + cÞ5ab + ac and for power over multiplication we have ða×bÞn 5an ×bn .
7 Conditional Statements Consider again the following statements: p: I win the lottery. q: I will buy you dinner. ” If I win the lottery and I buy you dinner, then I have kept my word. Thus, p → q is true when p is true and q is true. However, if I did not win the lottery and do not buy you dinner, or I did not win the lottery but do take you to dinner, I did not break my word. Hence, p → q is true when p is false whether q is true or false. Only when I win the lottery and do not take you to dinner has my word been broken.
Solution Constructing the truth table for the statement ∼ ðp ∧ qÞ ∨ p, we have p q pŸq ~ pŸq ~ pŸq ⁄ p T T T F T T F F T T F T F T T F F F T T Since the truth table for the statement ∼ ðp ∧qÞ ∨p is true, T, for all possible truth value combinations of p and q, the statement is a tautology. Analogous to tautologies, there are statements that are logically false; for example, p ∧ ∼ p is always false. If p is true then we have T ∧F, which is false; however, if p is false then we have F ∧T, which is false.