hw2 - (b If n is prime then it is odd(c For n to be...

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Introduction to Mathematical Reasoning Fall 2009 Assignment #2: Due Wednesday, 10/14/09 Part I: 1. Eccles, pp. 19–20, Exercises 2.1, 2.4, 2.5(ii). 2. Eccles, p. 53, Problems 2 and 3. 3. A statement form is called a tautology if it is true regardless of the truth values of its individual statement variables, and a contradiction if it is false regardless of their truth values. For example, the following truth table proves that the statement form P ∨∼ P is a tautology: P P P ∨ ∼ P T F T F T T Write out the truth table for each of the following statement forms, and determine if it is a tautology, a contradiction, or neither. (a) P ⇒ ∼ ( Q ( P )). (b) P (( R ) Q ) R . (c) ( P ( Q ( Q ))) P . 4. Consider the following implications: (a) n is prime only if it is odd.
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Unformatted text preview: (b) If n is prime, then it is odd. (c) For n to be composite, a necessary condition is that it not be even. (d) If n is even, then either it is composite or it is equal to 2. (e) If n is equal to 4, then it is neither prime nor odd. (f) n is odd if it is prime and not equal to 2. For each implication, do the following: (i) Determine the hypothesis and the conclusion. (ii) Translate it into a symbolic statement. (iii) Write its negation in symbolic form, and simplify it. (iv) Translate the negation back into an English statement. Use the abbreviations P ( n ), C ( n ), E ( n ), and O ( n ) with the same meanings as in Problem 4 of Assignment 1....
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