Order Discrete Math Assignment

The University of Maine at Augusta Name: ________________________ Mathematics Department Date: ________________________ MAT 280 F16

MAT 280 Exam 2 Proofs

Please answer the following questions. You my use your book and other outside resources.

However, all work should represent the individual student’s efforts. This exam covers

primarily sections 1.7 and 1.8, however there will be some questions from 1.5 and 1.6.

1. Show that the premises, “ Arron is registered as an instate student” and “All instate students have permanent addresses in Maine” imply the conclusion “Arron’s permanent address is in Maine”

2. Determine whether the following argument is valid.

Rainy days make gardens grow.

Gardens don’t grow if it is not hot.

It always rains on a day that is not hot.

Therefore, if it is not hot, then it is hot.

3. What is the rule of inference used in the following:

If I work all week on this take home exam, then I can answer the questions correctly. If I

answer the questions correctly, I will pass the exam. Therefore, if I work all week on this take

home exam, then I will pass the exam.

4. Suppose you wish to prove a theorem of the form “if p then q”.

a. If you give a direct proof, what do you assume and what do you prove?

b. If you give an indirect proof, what do you assume and what do you prove?

c. If you give a proof by contradiction, what do you assume and what do you prove?

5. Suppose you are allowed to give either a direct proof or a proof by contraposition of the following: if 5n 3 is even, then n is odd.

a. Which type of proof would be easier to give, direct or indirect? Explain why.

b. Prove: If 5n 3 is even then n is odd.

6. In the following proof, name the proof technique:

Prove that p→q

Proof: Assume that p → q is true….. Proof continued…..

7. Prove: If x is odd, then 7 x – 4 is odd

Proof:

1. Suppose x is odd. [Show 7x – 4 is odd .]

2. If x is odd then by definition, x = 2k 1, where k is an integer

So…

⇒ 7x – 4 = 7( 2k 1) – 4 (substitution) = 14k 7 – 4

= 14k 3

=2(7k 1) 1

= 2m 1 is odd for integer m =7k 1

Therefore, 7 x – 4 is odd.

Which of the following applies? a. method of direct proof d. proof by contrapositive

b. method of indirect proof c. proof by contradiction

e. b and d

8. What is wrong with the following “proof” that -2 = 2, using backward reasoning?

Assume that -2 = 2. (-2)2 = 22 Squaring both sides

4 = 4 Therefore -2 = 2.

9. It is important to ask a key question when attempting to prove a statement. For the key question, “How can I show that 2 lines in a plane are perpendicular?” which of the following answer(s) is/are incorrect? Explain why.

a. Show that the product of the slopes of the two lines equal -1.

c. Show that the lines are on adjacent sides of a rectangle. d. None

b. Show that each of the lines is perpendicular to a third line.

10. In class we mentioned that Key questions help to give oneself direction when starting a proof.

Suppose you are asked to prove, “If a, b, and c are integers and ab and bc then ac”.

The symbol means divides.

a. State a possible Key question that would help you to work through this proof.

b. Prove: If a, b, and c are integers and ab and bc then ac

11. Show that the statement ”The product of 2 irrational numbers is irrational” is false by finding a counterexample.

12. Given a class of 25 students prove that at least 4 of them were born on the same day of the week.

13. Give a direct proof of the following: “If m and n have different parities, then the sum of the m and n is odd”.

14. Give a proof by contradiction of the following: “If b is an odd integer, then b2 is odd”.

15. Prove: If a, b and c are even integers, then abc is divisible by 8.

16. Prove: If x and y are rational numbers then x divided by y is a rational number.

17. Prove or disprove that if m is an integer then 12

)3)(2)(1( mmmm is an integer.

18. Prove: If a and b are integers with a ≠ 0 and x is a real number such that ax2 bx b – a = 0, then a│b.

Note: a│b means that a divides b or a is a factor of b.

19. Given that a and b are integers and a ≠ 0. Prove that if a|300 and a|(b 300), then a|b. (see number 14 for definition of a|b.)

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