Discrete Math 2 Use the rules of inference to prove p s give

Discrete Math

2. Use the rules of inference to prove p s, given the following premises. Write your solution as a numbered sequence of statements. Identify each statement as either a premise, or a conclusion that follows according to a rule of inference from previous statements. In that case, state the rule of inference and refer by number to the previous statements that the rule of inference used.

(1) ¬r

(2) s

(3) q r

(4) q p

3. Prove or disprove the following statements.

(a) There exists a real number x such that x 2 > x + 9.

(b) For all integer x 6= 1 there exists an integer y such that xy = 6x + y.

4. Let m be an odd integer and n be an even integer. Prove that their sum is odd.

5. Use proof by contraposition to show that if mn is even then m is even or n is even.

Solution

4. let m be of the form 2a+1, which is odd, where a belongs to N.

    Let n be of the form 2b, which is even, where b belongs to N.

m + n = 2a+1 + 2b = 2(a+b) + 1

2 multiplied by any number is even.

m + n = even + 1 = odd.

Hence, sum of odd and even is odd.