Mth 110
Chapter 2: The Logic of Compound
Statements
August 16, 2011
Outline
1 2.1 Logical Form and Logical Equivalence
2 2.2 Conditional Statements
3 2.3 Valid and Invalid Arguments
4 2.4 Application: Digital Logic Circuits
• Central notion of deductive logic: argument form
• Argument: sequence of statements whose goal is to
establish the truth of an assertion
• The assertion at the end of the sequence is called the
conclusion while the preceding statements in an argument are called premises
The goal of an argument is to show that the truth of the conclusion follows necessarily from the truth of the premises.
Example
If x is a real number such that x < −5 or x > 5, then x 2 > 25.
Since x 2 > 25 then x < −5 and x > 5.
We can introduce the letters p, q, and r to represent statements that occur within our argument:
If p or q, then r .
Therefore, if not r , then not p and not q.
Example
Fill in the blanks in the argument (b) so that it has the same form as the argument (a). Then, write the common form of the argument using letters to replace the individual statements.
(a)
If it rains today or I have a lot of work to do, I won’t go for a walk. I have a lot of work to do.
Therefore, I won’t go for a walk.
(b)
If MTH110 is easy or
then
.
I will study hard.
Therefore, I will get an A in this course.
Solution:
1
I will study hard.
2
I will get an A in this course.
Common form of the arguments:
If p or q, then r . q Therefore, r .
Statements
Definition
A statement (or, proposition) is a sentence that is true or false but not both.
Examples
1
“The area of the circle of radius r is r 2 π” is a statement. So is “sin(π/2) = 0”. The first is a true statement, while the second one is false.
2
“x + 2 ≥ y ” is not a statement; namely, for some values of x and y , e.g. x = 1, y = 2, it is true, while for some other values (e.g. x = −1, y = 2), it is false.
Compound Statements
Statements
August 16, 2011
Outline
1 2.1 Logical Form and Logical Equivalence
2 2.2 Conditional Statements
3 2.3 Valid and Invalid Arguments
4 2.4 Application: Digital Logic Circuits
• Central notion of deductive logic: argument form
• Argument: sequence of statements whose goal is to
establish the truth of an assertion
• The assertion at the end of the sequence is called the
conclusion while the preceding statements in an argument are called premises
The goal of an argument is to show that the truth of the conclusion follows necessarily from the truth of the premises.
Example
If x is a real number such that x < −5 or x > 5, then x 2 > 25.
Since x 2 > 25 then x < −5 and x > 5.
We can introduce the letters p, q, and r to represent statements that occur within our argument:
If p or q, then r .
Therefore, if not r , then not p and not q.
Example
Fill in the blanks in the argument (b) so that it has the same form as the argument (a). Then, write the common form of the argument using letters to replace the individual statements.
(a)
If it rains today or I have a lot of work to do, I won’t go for a walk. I have a lot of work to do.
Therefore, I won’t go for a walk.
(b)
If MTH110 is easy or
then
.
I will study hard.
Therefore, I will get an A in this course.
Solution:
1
I will study hard.
2
I will get an A in this course.
Common form of the arguments:
If p or q, then r . q Therefore, r .
Statements
Definition
A statement (or, proposition) is a sentence that is true or false but not both.
Examples
1
“The area of the circle of radius r is r 2 π” is a statement. So is “sin(π/2) = 0”. The first is a true statement, while the second one is false.
2
“x + 2 ≥ y ” is not a statement; namely, for some values of x and y , e.g. x = 1, y = 2, it is true, while for some other values (e.g. x = −1, y = 2), it is false.
Compound Statements