Classify each of the following statements as true, false, or not a valid

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TRUE OR FALSE

1. 1.     “p and q” is false if both p and q are false.
2. 2.     If “p and q” is false, then both p and q are false.
3. 3.     It is possible for both “p and q” and “p or q” to be false.
4. 4.     It is possible for both “p and q” and “p or q” to be true.
5. 5.     The implication “if =5, then =9” is true.

Exercises

**I just need help with (d) and (e) for this question.

1. 1.     Classify each of the following statements as true, false, or not a valid mathematical statement.

(a)    [BB} An integer is a rational number.

(b)    [BB] let x denote a real number.

(c)    [BB} The square of a real number is a positive number.

(d)    Where is Newfoundland?

(e)    The product of an integer and an even integer is an even integer.

      3.  Rewrite the each of the following statement so that it is clear that each is an implication.

              (b) The product of rational numbers is rational.

     4. Determine whether each of the following implications is true or false

              (b) If 7 is odd, then 25 is odd.

    5. Write down the negation of each of the following statements in clear and concise English. Do not use

         the expression “It is not the case that” in your answers.

              (a) [BB] Either >0 or  is not a real number.

              (b) is a real number and +1 = 0.

6. Write down the converse and the contrapositive of each of the following implications.

             (d) If  is odd integer, then is an even integer.

7. Rewrite each of the following statements using the quantifiers “for all” (or “for every”) and “there exsist” as appropriate.

             (f) All positive real numbers have real square roots.

             (h) There is no smallest positive real number.

1. Construct a truth table for each of the following compound statements.

             (b)  (p   q)  ((p) q)

             (c)   (p   (q  p))  p

3. [BB] Determine the truth value for:

      [p  (q  ())]  [(()  q)]