In the previous post, I have promised that we will take our discussion about mathematical proofs to the next level. We will start this journey by learning about propositions.
In our daily lives, we often encounter statements that are either true or false. Some examples are shown below.
1.) Australia is an island continent.
2.) The Earth revolves around the sun.
3.) Asia is the largest continent.
4.) The sum of 3 and 5 is 9.
5.) The Earth has two moons.
As we can see, the first three statements above are true while the last two statements are false.
Statements that are either true or false, but not both are called propositions. If a proposition is true, then its truth value is True (usually denoted by T) and if it is false, its truth value is False (usually denoted by F). Another characteristic of a proposition is it is a declarative sentence. Therefore, to check if a statement is a proposition or not, you have to check if (1) it is a declarative sentence and (2) it is either true or false.
Exercise: Determine of the statement is a proposition or not. If a statement is a proposition, determine its truth value. You can read the answer key below.
1.) Welcome to Jamaica!
2.) 2 is a prime number.
3.) Neil Armstrong was the first human to step on the moon.
4.) A square has four vertices.
5.) There are 1000 prime numbers in all.
Propositions are usually represented by letters, sometimes capital and sometimes small. In this series, we will use small letters to represent propositions. Hence, if we let p be the fourth proposition above, we can write this proposition as follows.
p: A square has four vertices.
The advantage of using letters is that we can refer to them in discussions rather than repeating the entire statement. In the following posts, you will see why this representation is convenient.
In the next post, we will be discussing compound propositions.
Answer to the Exercise above.
1.) Not a proposition
2.) Proposition, true
3.) Proposition, true
4.) Proposition, true
5.) Proposition, false