Problem
1.
Determine whether the following two statements are equivalent:
(p Ú q) Ú (p Ù r) and (p Ú q) Ù r
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
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
Let
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 6^{th} 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)
Problem
5.
Find decimal representation for the following binary number: 00001001
0001001 = 1 * 2^{0} + 1 * 2^{3} = 9
Problem
6.
Write a negation to the following statement:
The sum of any two irrational numbers is irrational
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
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