Discrete Math Test #2 Study Guide Name: ___________________
Show all step. Explain your logic. Use the appropriate mathematical language.
1. Consider the following statement:
Statement A: ∀ integers m and n, if 2m + n is even then m and n are both even.
a) Write a negation for Statement A.
(b) Disprove Statement A. That is, show that Statement A is false.
2.
If m and n are integers, is 10m + 6n +1 an even integer? Justify your answer.
3. Prove the following statement directly from the definitions of the terms. Do not use any other facts previously proved in class or in the text or in the exercises.
For all integers a, and b, if a divides b then a divides b.
4. Prove the statement below directly from the definitions of the terms. Do not use any other facts previously proved in class or in the text or in the exercises.
The sum of any three consecutive integers can be written in the form 6n + 3 for some integer n.
5. Prove the following statement by contradiction: For all real numbers x and y, if y is irrational and x is rational, then x+ y is irrational.
6.
Consider the following statement: For all real numbers r, if r is irrational then r is irrational.
a) Prove the statement by contradiction.
b) Prove the statement by contraposition.
7.
Prove that is irrational.
8.
Prove by contradiction that 7 + 2 is irrational. You may use the fact that is irrational.
9.
Use mathematical induction to prove that for all integers n ≥ 3, 3 + 4 + 5 + · · · + n =.
10. Use mathematical induction to prove that for all integers n ≥ 5, 1 + 4n < 2n.
11. Use a truth table to see if the argument is valid: