Math 2345 Test #1

1. Give an example of a statement that is a tautology. Use English or logical symbols.

2. a) Make a truth table for the statement (~pq)~q.

b) If p is true and q is false, is the statement (~pq)~q true or false?

3. Give the converse of the statement: If Fyodor is strong, then Fyodor is fearful.

4. Negate the statements. Write your answers in good English:

a) Joe eats burgers and Jane does not like fish.

b) If Sara is a survivor, then Sara is famous.

c) Every hockey player is missing some teeth.

5. Rewrite the statement in good English with no mathematical symbols:

real numbers x, zZ such that z>x.

6. Rewrite the statement formally using quantifiers and variables:

Every cloud has a silver lining.

7. Give a counterexample to show that the statement is false:

If any 2 even integers are added together, then the sum of the 2 even integers is divisible by 4.

8. Write the first 4 terms of the sequence defined by                

9. Write the sum using summation notation:      .

10. Compute the sum and simplify:            

11. Compute the product and simplify     : .

12: Simplify:     , assuming n>0.

13. Give the contrapositive of the following statement:

For every integer n, if n is even then is divisible by 4.

14. Give a direct proof of the statement: If n is any even integer, then .

15. Give a direct proof of the statement: For all integers a, b, and c, if a|b and a|c, then a|(b-c).

16. Prove the statement using the method of mathematical induction:

1 + 3 + 5 + + (2n-1) = for all integers n 1.

17. Prove the following statement either by contraposition or by contradiction. Indicate your choice of method:

For all integers n, if is odd, then n is odd.