# Discrete Mathematics, Summer 2002

## Quiz #1

Problem 1.

Determine whether the following two statements are equivalent:

(p Ú q) Ú (p Ù r) and (p Ú q) Ù r

# Solution

We will use a truth table to determine the equivalence of two expressions:

 p q r p Ú q p Ù r (p Ú q) Ú (p Ù r) (p Ú q) Ù r F F F F F F F F F T F F F F F T F T F T F F T T T F T T T F F T F T F T F T T T T T T T F T F T F T T T T T T T

Comparing last two columns, we can say that two given expressions are not equivalent

Problem 2.

Suppose that p and q are statements so that p à q is false. Find the values for the following two statements:

a)      p Ú q

b)      q à p

### Solution

By definition of a conditional operator à, p à q being false means that p is true and q is false. Having determined that, we obtain:

a)      p Ú q is true, since p is true

b)      q à p is true, since q is false

Problem 3.

Determine the validity of the following argument.

Premises:

Oleg is a math major or Oleg is an economics major.

If Oleg is a math major, then Oleg is required to take Math 362

Conclusion:

Oleg is an economics major or Oleg is not required to take Math 362

### Solution

Let

• p = Oleg is a math major
• q = Oleg is an economics major
• r = Oleg is required to take Math 362

This way, we have two premises: p Ú q and p à r. We also have a conclusion in the form q Ú ~r. To identify the validity of the argument we build truth table:

p

q

r

p Ú q

p à r

q Ú ~r

F

F

F

F

T

T

F

F

T

F

T

F

F

T

F

T

T

T

F

T

T

T

T

T

T

F

F

T

F

T

T

F

T

T

#### T

F

T

T

F

T

F

T

T

T

T

T

T

T

Critical rows are bold and italicized. The third critical row (which is the 6th row in the entire table) shows that the argument is invalid.

Problem 4.

Construct a circuit for the following Boolean expression: (P Ù ~Q) Ú (~P Ù R)

### Solution

Problem 5.

Find decimal representation for the following binary number: 00001001

### Solution

0001001 = 1 * 20 + 1 * 23 = 9

Problem 6.

Write a negation to the following statement:

The sum of any two irrational numbers is irrational

### Solution

There are two irrational numbers whose sum is not irrational

Problem 7.

Write converse, inverse and contrapositive to the following statement:

If the square of an integer is even, then the integer is even

### Solution

Contrapositive: if an integer is odd, then its square is odd

Converse: if an integer is even, then its square is even

Inverse: if the square of an integer is odd, then the integer is odd