Exam 1

100 points possible.

(As usual, don't write trivial negations, like ``It is not the case that blah blah.'' Show me you have learned something.)

1. (5 pts.) Write the negation for the following statement.

The bus was late and logic is not confusing.

2. (5 pts.) Arguments I and II are supposed to have the same logical form, but Argument II has some blanks.

Argument I:

This number is even or this number is odd.

This number is not even.

Therefore, this number is odd.

Argument II:

or interest rates are steady.

Stocks are not increasing.

Therefore,
.

(a) Represent the logical form of Argument I using letters to stand for the component sentences.

(b) Fill in the blanks so that Argument II has the same logical form as Argument I. (You don't have to write the whole argument, just what goes in the blanks.)

3. (5 pts.) Use a truth table to establish that a conditional statement is not logically equivalent to its inverse.

4. (5 pts.) Rewrite the following statement in if-then form.

A sufficient condition for Jack to fall down is that Humpty Dumpty fall down.

5. (5 pts.) Use a truth table to determine whether the argument form is valid or invalid. Clearly label the ``critical rows.''

p

      Therefore

pÚq

Any valid argument with true premises has a true conclusion.

7. (5 pts.) Write the negation of the following statement.

"x Î R, if sin x > 0, then sec x > 0 or tan x < 0.

8. (5 pts.) Consider the following statement.

Everybody is older than somebody.

(a) Rewrite the given statement formally using quantifiers and variables.

(b) Write a negation for the statement, using quantifiers and variables.

9. (5 pts.) Consider the following statement.

Every action has an equal and opposite reaction.

(a) Rewrite that statement formally using quantifiers and variables.

(b) Write a negation for the statement, using quantifiers and variables.

10. (5 pts.) State whether the following argument has a valid or invalid form. (You do not have to justify your answer.)

All cheaters sit in the back row.

      Therefore

George is not a cheater.

11. (5 pts.) The given statement is true. Write the beginning of a proof (just the beginning, not the whole proof). Include the starting point and what is to be shown.

For all integers m, if m > 1, then 0 < 1/m < 1.

12. (10 pts.) Is the given statement true or false? If it is true, prove it. If it is false, give a counterexample.

If a sum of two integers is even, then one of the summands is even. (Note: In the expression a+b, the summands are a and b.)

13. (10 pts.) The given statement is true. Prove it directly from the definitions of odd and even.

If n is an odd integer, then n²+3n is even.

14. (5 pts.) Let x=-18.01.

(a) Compute ëxû.

(b) Compute éxù.

15. (10 pts.) For each of the values of n and d, find integers q and r such that both n=dq+r and 0 £ r < d.

(a) n=50,       d=65

(b) n=-26,       d=11

16. (10 pts.) The given statement is true. Prove it.

The product of any four consecutive integers is divisible by 4.