Basic Logical Operations

Basic Logical Operations

Logical operations are used to form new propositions by combining or modifying existing ones. These operations are fundamental in reasoning, computer science, and digital logic.

1. Negation (NOT)

The negation of a proposition $p$, written as $\neg p$, is true when $p$ is false, and false when $p$ is true. It simply reverses the truth value.

Example:
Let $p$: "It is raining."
Then $\neg p$: "It is not raining."

2. Conjunction (AND)

The conjunction of two propositions $p$ and $q$, written $p \land q$, is true only when both $p$ and $q$ are true.

Example:
$p$: "It is sunny."
$q$: "It is warm."
$p \land q$: "It is sunny and warm."

3. Disjunction (OR)

The disjunction $p \lor q$ is true if at least one of $p$ or $q$ is true. This is the inclusive OR.

Example:
$p$: "I will study."
$q$: "I will play."
$p \lor q$: "I will study or play (or both)."

4. Exclusive OR (XOR)

The exclusive OR $p \oplus q$ is true if exactly one of $p$ or $q$ is true, but not both.

Example:
Two switches controlling a light—only one switch can be ON at a time for the light to be ON.

5. Implication (IF...THEN)

The implication $p \rightarrow q$ means "if $p$, then $q$." It is false only when $p$ is true and $q$ is false.

Example:
$p$: "It rains."
$q$: "The ground gets wet."
"If it rains, then the ground gets wet."

Even if it doesn’t rain (p is false), the implication is still considered true, regardless of whether the ground gets wet or not.

6. Biconditional (IF AND ONLY IF)

The biconditional $p \leftrightarrow q$ is true when both $p$ and $q$ are either true or false (i.e., have the same truth value).

Example:
$p$: "You can vote."
$q$: "You are 18 or older."
"You can vote if and only if you are 18 or older."

These operations form the foundation for logical arguments, digital circuits, and programming decision-making.