Logical Operators

Definition

Logical operators combine or modify propositions.

They create new propositions from existing ones.

Examples:

P AND Q
P OR Q
NOT P
If P, then Q

Logical operators build compound propositions.

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Why Operators Matter

A single proposition can say one thing.

Example:

P = "It is raining"

Logical operators let us build more structured statements.

Example:

P AND Q

This says both P and Q hold.

Another example:

NOT P

This says P does not hold.

Operators let reasoning express relationships between statements.

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Operators Use Truth Values

Logical operators work on truth values.

Example:

P = true
Q = false

Then:

P AND Q = false
P OR Q = true
NOT P = false

The operator defines how the output truth value is produced.

A logical operator is a rule for truth values.

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Common Operators

The basic operators are:

AND
OR
NOT
IMPLIES
EQUIVALENT

Each operator has a specific rule.

Example:

AND is true only when both parts are true.

Example:

OR is true when at least one part is true.

Each logical operator defines a different truth pattern.

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Operators Create Structure

Logical operators can be nested.

Example:

(P AND Q) OR R

This expression has structure:

first combine P and Q
then combine the result with R

Parentheses help make the structure clear.

Logical operators build larger statements from smaller statements.

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Final Summary

• Logical operators combine or modify propositions.
• They produce new propositions.
• They operate on truth values.
• Common operators include AND, OR, NOT, implication, and equivalence.
• Operators define truth patterns.
• Operators allow complex reasoning structures to be built from simple statements.