Question

1. Construct a truth table for ‘‘P does not imply Q’’. 2. Let P, Q be logical statements. Reformulate the statement ‘‘P does not imply Q’’ using only conjunction and negation. 3. Let P(x), Q(x) be logical statements with a real parameter x. Reformulate the statement ‘‘For every x ? ?, P(x) does not imply Q(x)’’ using only quantifiers, conjunction, and negation. 4. There are two different ways to put parentheses in the expression P ? Q ? R. Are they equivalent? If yes, construct truth tables. If not, explain why not. 5. P ? Q ? R can also be read as ‘‘(P ? Q) and (Q ? R)’’. Is this equivalent to any of the other two interpretations? 6. Let P(x), Q(x) be logical statements with a parameter x ? S. Is ‘‘?x ? S, P(x) and Q(x)’’ the same as ‘‘(?x ? S, P(x)) and (?x ? S, Q(x))’’? What about ‘‘?x ? S, P(x) or Q(x)’’ versus ‘‘(?x ? S, P(x)) or (?x ? S, Q(x))’’? Here S is an arbitrary scope. You may consider the case S = {0, 1} if you find this case easier. 7. Let an be a sequence of real numbers, and let a be a real number. The following is the definition of convergence of an to a. For any real number ? > 0, there exists a natural number N such that for any natural number n > N we have |an - a| < ?. Write the negation of this, i.e. the definition of non-convergence of an to a. Your definition should only use positive statements, which means that you cannot use ‘‘not’’. 8. What does the statement ?x, y ? ?, x < y ? (?z ? ? s.t. x < z < y) mean?

          1. Construct a truth table for ‘‘P does not imply Q’’.
2. Let P, Q be logical statements. Reformulate the statement ‘‘P does not imply Q’’ using only conjunction and negation.
3. Let P(x), Q(x) be logical statements with a real parameter x. Reformulate the statement ‘‘For every x ? ?, P(x) does not imply Q(x)’’ using only quantifiers, conjunction, and negation.
4. There are two different ways to put parentheses in the expression P ? Q ? R. Are they equivalent? If yes, construct truth tables. If not, explain why not.
5. P ? Q ? R can also be read as ‘‘(P ? Q) and (Q ? R)’’. Is this equivalent to any of the other two interpretations?
6. Let P(x), Q(x) be logical statements with a parameter x ? S. Is ‘‘?x ? S, P(x) and Q(x)’’ the same as ‘‘(?x ? S, P(x)) and (?x ? S, Q(x))’’? What about ‘‘?x ? S, P(x) or Q(x)’’ versus ‘‘(?x ? S, P(x)) or (?x ? S, Q(x))’’? Here S is an arbitrary scope. You may consider the case S = {0, 1} if you find this case easier.
7. Let an be a sequence of real numbers, and let a be a real number. The following is the definition of convergence of an to a. For any real number ? > 0, there exists a natural number N such that for any natural number n > N we have |an - a| < ?. Write the negation of this, i.e. the definition of non-convergence of an to a. Your definition should only use positive statements, which means that you cannot use ‘‘not’’.
8. What does the statement ?x, y ? ?, x < y ? (?z ? ? s.t. x < z < y) mean?

1. Construct a truth table for ‘‘P does not imply Q’’.
2. Let P, Q be logical statements. Reformulate the statement ‘‘P does not imply Q’’ using only conjunction and negation.
3. Let P(x), Q(x) be logical statements with a real parameter x. Reformulate the statement ‘‘For every x ? ?, P(x) does not imply Q(x)’’ using only quantifiers, conjunction, and negation.
4. There are two different ways to put parentheses in the expression P ? Q ? R. Are they equivalent? If yes, construct truth tables. If not, explain why not.
5. P ? Q ? R can also be read as ‘‘(P ? Q) and (Q ? R)’’. Is this equivalent to any of the other two interpretations?
6. Let P(x), Q(x) be logical statements with a parameter x ? S. Is ‘‘?x ? S, P(x) and Q(x)’’ the same as ‘‘(?x ? S, P(x)) and (?x ? S, Q(x))’’? What about ‘‘?x ? S, P(x) or Q(x)’’ versus ‘‘(?x ? S, P(x)) or (?x ? S, Q(x))’’? Here S is an arbitrary scope. You may consider the case S = 0, 1 if you find this case easier.
7. Let an be a sequence of real numbers, and let a be a real number. The following is the definition of convergence of an to a. For any real number ? > 0, there exists a natural number N such that for any natural number n > N we have |an - a| < ?. Write the negation of this, i.e. the definition of non-convergence of an to a. Your definition should only use positive statements, which means that you cannot use ‘‘not’’.
8. What does the statement ?x, y ? ?, x < y ? (?z ? ? s.t. x < z < y) mean?

Added by Bradley P.

Question

Please give Ace some feedback