# Propositional Logic

## Introduction to Logic

Logic defines a formal language for representing knowledge and for making logical inferences. It helps us to understand how to construct a valid argument.
Logic defines:
• Syntax of statements
• The meaning of statements
• The rules of logical inference (manipulation)

### Propositional Logic

Definition: A proposition is a statement that is either true or false.
Examples:
• Dubai is the capital city of UAE. Answer: False.
• 3 + 4 = 7. Answer: True.
• How are you today? Answer:A question is not a proposition.
• x + y = 3 Answer: It is neither true nor false.

### Composite Statements

More complex propositional statements can be build from elementary statements using logical connectives.
Examples:
• Proposition A: ″ It is sunny today″
• Proposition B: ″We will go to the park″
• A new (combined) proposition:″If it is sunny today then we will go to the park″

### Logical connectives

• Negation
• Conjunction
• Disjunction
• Exclusive
• Implication
• Biconditional

#### Negation

Definition: Let p be a proposition. The statement ″It is not the case that p.″ is another proposition, called the negation of p. The negation of p is denoted by ¬p and read as "not p″.
Examples
• Pasadena is located in the State of California. The negation of this statement is:
It is not the casethat Pasadena is located in the State of California.
• 15 is not a prime number.
Exercise: Negate the following propositions:
• It is sunny outside. Answer: It is not sunny outside
• 3 is a prime number. Answer: 3 is not a prime number
• Spain is in Europe. Answer: Spain is not in Europe.