Discrete Structures Exam4

Question 1. Use a truth table to show that the following wff is a tautology.

Question 2. Prove the following wff with propositional logic.

Question 3.  Give a counterexample to disprove the following statement.

For every prime number n, n + 4 is prime.

(Recall that a prime number is an integer n > 1 such that n is not divisible by any integers other than 1 and n.)

Question 4. Write a direct proof for the following statement.

The sum of three consecutive integers is divisible by 3.

Question 5. Write a proof by contradiction for the following statement.

If p and q are primes and p divides q, then p = q.

(Recall that a prime number is an integer n > 1 such that n is not divisible by any integers other than 1 and n.)

Question 6. Write a proof by contraposition for the following statement.

If x + y > 100, then x > 50 or y > 50.

(Recall that and ).

Question 7. Use the First Principle of Mathematical Induction to prove the following statement for every positive integer n.