DISCRETE MATH: FINAL REVIEW

[Pages:26]DISCRETE MATH: FINAL REVIEW

DR. DANIEL FREEMAN

1. Chapter 1 review 1) a. Does 3 = {3}?

b. Is 3 {3}? c. Is 3 {3}? c. Is {3} {3}? c. Is {3} {3}? d. Does {3} = {3, 3, 3, 3}? e. Is {x Z|x > 0} {x R|x > 0}?

2. Chapter 2 review 1) Construct a truth table for ( p q) (p q).

2) Construct a truth table to show that (p q) is logically equivalent to p q. What is the name for this law?

1

2

DR. DANIEL FREEMAN

3) Construct a truth table to determine if (p q) r is logically equivalent to (p r) (q r).

4) Use a truth table to determine if the following argument is logically valid. Write a sentence which justifies your conclusion.

pqr r pq pr

DISCRETE MATH: FINAL REVIEW

3

You will be provided with the following information on the test.

2.1. Modus Ponens and Modus Tollens. ? The modus ponens argument form has the following form: If p then q. p q. ? Modus tollens has the following form: If p then q. q p.

2.2. Additional Valid Argument Forms: Rules of Inference.

? A rule of inference is a form of argument that is valid. Modus ponens and

modus tollens are both rules of inference. Here are some more...

Generalization

p Elimination p q

pq

q

Specialization

pq

p

p Transitivity p q

Proof by Division into Cases p q

qr

pq

pr

q r Conjunction p

r

q

Contradiction Rule

pc

pq

p

4

DR. DANIEL FREEMAN

5) Write a logical argument which determines what I ate for dinner. Number each step in your argument and cite which rule you use for each step.

a. I did not have a coupon for buns or I did not have a coupon for hamburger. b. I had hamburgers or chicken for dinner. c. If I had hamburgers for dinner then I bought buns. d. If I did not have a coupon for buns then I did not buy buns. e. If I did not have a coupon for hamburger then I did not buy buns.

DISCRETE MATH: FINAL REVIEW

5

3. Chapter 3 review 1) a. Give an example of a universal conditional statement.

b. Write the contrapositive of the example.

c. Write the negation of the example.

2) Write the following statements symbolically using , , , , . Then write their negation.

a. If x, y R then xy + 1 R.

b. Every nonzero real number x has a multiplicative inverse y.

c. Being divisible by 8 is not a necessary condition for an integer to be divisible by 4.

d. If I studied hard then I will pass the test.

6

DR. DANIEL FREEMAN

2) Is it true or false that every real number bigger than 4 and less than 3 must be negative? Explain why.

4. Chapter 4 review 1) Prove using the definition of odd: For all integers n, if n is odd then (-1)n = -1.

2) Prove using the definition of even: The product of any two even integers is divisible by 4.

DISCRETE MATH: FINAL REVIEW

7

3) Prove: For each integer n with 1 n 5, n2 - n + 11 is prime.

4) Show that: .123123123... is a rational number.

5) Prove using the definition of divides: For all integers a, b, and c, if a divides b and a divides c then a divides b - c.

8

DR. DANIEL FREEMAN

6) Evaluate 60 div 8 and 60 mod 8.

7) Prove using the definition of mod: For every integer p, if pmod10 = 8 then pmod5 = 3.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download