Chapter 1: Proposition

Fundamentals of Mathematical Logic

Logic is commonly known as the science of reasoning.  The emphasis here will be on logic as a working tool.  Some of the reasons to study logic are the following:

• At the hardware level the design of 'logic' circuits to implement instructions is greatly simplifies by the use of symbolic logic.
• At the software level a knowledge of symbolic logic is helpful in the design of programs.

Proposition

A proposition is any meaningful statement that is either true or false, but not both.  We will use lowercase letters such as p, q, r,... to represent propositions.  The truth value of a proposition is true denoted by T, if it is true statement and false, denoted by F, if it is a false statement.  Statements that are not propositions include questions and commands.

Example 1.1:

Which of the following are propositions?  Give the truth value of the propositions.

• 2 + 3 = 7
• Tun Mahathir was Prime Minister of Malaysia.
• What time is it?
• Keep quiet!

Solution

• A proposition with Truth Value (F).
• A proposition with Truth Value (T)
• Not a proposition since no truth value can be assigned to this statement.
• Not a proposition since this is a command.

Example 1.2:

Which of the following are proposition?  Give the truth value of the propositions.

• The difference of two primes.
• 2 + 2 = 4
• Kuching is the capital of Malaysia
• How old are you?

 Solution

• Not a proposition
• A proposition with Truth Value (T)
• A proposition with Truth Value (F)
• Not a proposition