Negation and the Not Connector

In the previous post, we have discussed compound propositions. We learned that compound proposition is a proposition formed from simple propositions using some logical connectors. The first logical connector that we are going learn is about “not” which is used for negation.

The negation of a proposition p denoted by ~p (read as “not p”) and is defined by the following truth table. As we can see, if p is true, then ~p is false. If p is false, then ~p is true.

Consider the following negations.

Example 1
q: I am going to the party.
~q: I am not going to the party.

Example 2
r: John is wealthy.
~r: John is not wealthy.  Continue reading…

 

Introduction to Compound Propositions

In the previous post, we have learned about propositions. We learned that propositions are statements that are either true or false but not both. In this post, we are going to combine two or more propositions using words such as and, or, and if and then. Two or more propositions combined are called compound propositions and the words used to combined them are called logical connectors. We formalize our knowledge about compound propositions by the following definition.

Definition

A compound proposition is a proposition formed from simpler propositions using logical connectors or some combination of logical connectors. Some logical connectors involving propositions p and/or q may be expressed as follows: not p, p and q, p or q, if p then qContinue reading…

 

Understanding Propositions

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.  Continue reading…