What is a proposition?
A proposition is a statement that is either true or false.
Examples of propositions:
1. 2 + 3 = 5
2. 1 + 1 = 3
The first statement " 2 + 3 = 5 " is a proposition whose value is true.
The second statement " 1 + 1 = 3 " is a proposition whose value is false since 1 + 1 is not equal to 3.
Slide 2
Statements that are not proposition
1. X + Y > 4
2. X = 3
3. Are you leaving?
4. Buy four books
X and Y are variables. So, any operation on variables cannot be a proposition, since we don't know if the statement is true or false.
A question is also not a proposition, since it does not have a true or false value.
Similarly, orders such as "Buy four books" is not a proposition as it does not have a true or false value.
Slide 3
Propositional Variables
Propositional logic is the study of how propositions are combined and related.
A propositional variable is a letter such as P, Q, R that represents a proposition. For example, P is the proposition: All birds can fly.
Propositional variables can have only the values: True or False.
Such variables are also called Boolean variables.
Slide 4
Compound Propositions
We can combine, modify and relate propositions using words such "not", "and", "or", "implies" and "if-then". For example, we can combine three propositions like this:
If all humans are mortal and all Greeks are human, then all Greeks are mortal.
Here, there are three propositions which have been combined to form a compound proposition.
Let's denote each proposition with a propositional variable.
A: All humans are mortal
B: All Greeks are human
C: All Greeks are mortal.
The compound proposition can then be written as:
If A and B, then C.
Slide 5
Logical Connectives
The words connecting propositions such as "and", "or", "if-then" are called logical connectives.
There are several logical connectives.
NOT, AND, OR, IF-THEN, IMPLIES, IF-AND-ONLY-IF
First, we shall study the three logical connectives: "NOT", "AND" and "OR".
We can represent compound propositions using logical connectives. Examples are:
P AND Q
IF P THEN Q
