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Category > Management Posted 04 Feb 2018 My Price 9.00

mathematical logic

Which of the sentences are statements?

 

(a) If x is a real number, then x2 ≥ 0.

(b) Seven is a prime number.

(c) Seven is an even number.

(d) This sentence is false. In studying mathematical logic we shall not be concerned with the truth value of any particular simple statement. To be a statement, it must be either true or false (and not both), but it is immaterial which condition applies. What will be important is how the truth value of a compound statement is determined by the truth values of its simpler parts. In everyday English conversation we have a variety of ways to change or combine statements. A simple statement‡ like

Sentence (e) is known as the Gold bach conjecture after the Prussian mathematician Christian Goldbach, who made this conjecture in a letter to Leonhard Euler in 1742. Using computers it has been verified for all even numbers up to 1014 but has not yet been proved for every even number. For a good discussion of the history of this problem, see Hofstadter (1979). Recent results are reported in Deshouillers et al. (1998). ‡ It may be questioned whether or not the sentence “It is windy” is a statement, since the term “windy” is so vague. If we assume that “windy” is given a precise definition, then in a particular place at a particular time, “It is windy” will be a statement. It is customary to assume precise definitions when we use descriptive language in an example. This problem does not arise in a mathematical context because the definitions are precise.

The italicized words above (not, and, or, if . . . then, if and only if ) are called sentential connectives. Their use in mathematical writing is similar to (but not identical with) their everyday usage. To remove any possible ambiguity, we shall look carefully at each and specify its precise mathematical meaning. Let p stand for a given statement. Then ~ p (read not p) represents the logical opposite (negation) of p. When p is true, ~ p is false; when p is false, ~ p is true. This can be summarized in a truth table:

where T stands for true and F stands for false.

Answers

Maurice Tutor
(5)
Status NEW Posted 04 Feb 2018 07:02 PM My Price 9.00

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