Discrete Math problem

MAT 251 Module One Exam Summer 2017

Due Sunday, June 18 at 11:59 pm. This is a firm deadline, so please plan accordingly to finish

on time and make sure (in advance) that your access to a scanner is reliable.

Answer all questions on separate paper and show all work wherever possible. PLEASE USE

BLUE OR BLACK INK AND PLEASE MAKE SURE YOUR PRINTING IS DARK.

1. a. Using the symbols p: You play the game and q: You win

Express the sentence “If you don’t play the game, you don’t win” using symbolic

logic.

b. Find the contrapositive of the sentence in part a). Express your answer both in

words, and in symbolic form.

2. a. Find a counterexample to disprove the following statement:

For all 1n ≥ , 2

4 2 6m m− + is divisible by 4.

b. Find a counterexample to disprove the following statement:

2

, 1 9n n∀ ∈ − ≠ℕ

3. a. Find the negation of the following statement. Use symbolic notation.

, 2 1 0x x∀ ∈ + ≤ℝ .

b. Find the negation of the following statement. Use symbolic notation.

, 2 7 10m m∃ ∈ + =ℕ .

4. a. Use truth tables to prove that p q⇒ is logically equivalent to p q∨ .

b. Use truth tables to prove that ( )p q∨ is logically equivalent to p q∧ .

5. Complete a truth table for the following logical expression:

[ ] [ ]( ) ( ) ( )p q r p q p r∧ ∨ ⇔ ∧ ∨ ∧

6. Use a direct proof to prove the following:

Prove that the sum of any five consecutive odd integers is divisible by 5.

7. Use a proof by contrapositive to prove the following:

Let m and n be integers. Prove that if mn is even, then either m is even or n is even.

8. Use a proof by contradiction to prove the following:

Prove that 7 is irrational.

9. Use a proof by mathematical induction to prove the following:

Prove that for all 1n ≥ , 2

12 48 108 ... 12 2 ( 1)(2 1)n n n n+ + + + = + +

10. Use a proof by cases OR a proof by mathematical induction to prove the following:

Prove that for all 1n ≥ , 5

5n n− + is divisible by 5.