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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 (~p q) ~q.
b) If p is true and q is false, is the statement (~p q) ~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, z Z 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.