Discrete Mathematics, Summer 2002

 

Quiz #1

 

Problem 1.

 

Determine whether the following two statements are equivalent:

(p q) (p r) and (p q) r

 

Solution

 

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

 

p

q

r

p q

p r

(p q) (p r)

(p q) 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 q is false. Find the values for the following two statements:

a)      p q

b)      q p

 

Solution

 

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

a)      p q is true, since p is true

b)      q 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

 

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

 

p

q

r

p q

p r

q ~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 ~Q) (~P 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