Sure, let’s add examples for each logical operator with the propositions p, q, r, and s:
Negation (NOT)
The unary logical negation, represented by the exclamation mark (!), flips the truth value of a proposition. If the proposition is true, the negation makes it false, and if it’s false, the negation makes it true. For example:
| p | !p |
|---|---|
| true | false |
| false | true |
Disjunction (OR)
The OR operator, represented by two vertical bars (∨), evaluates to true if either or both of the operands are true. It returns false only if both operands are false. For example:
| p | q | p ∨ q |
|---|---|---|
| true | true | true |
| true | false | true |
| false | true | true |
| false | false | false |
Example:
- p: It’s raining today
- q: It’s sunny today
- p ∨ q: It’s either raining or sunny today
Exclusive OR (XOR)
The exclusive OR operator, represented by the caret (^), evaluates to true if and only if exactly one of the operands is true. It returns false if both operands are true or both are false. For example:
| p | q | p ^ q |
|---|---|---|
| true | true | false |
| true | false | true |
| false | true | true |
| false | false | false |
Example:
- p: The light is on
- q: The door is open
- p ^ q: Either the light is on or the door is open, but not both
Conjunction (AND)
The logical conditional binary AND operator, represented by two ampersands (∧), evaluates to true if both operands are true; otherwise, it returns false. For example:
| p | q | p ∧ q |
|---|---|---|
| true | true | true |
| true | false | false |
| false | true | false |
| false | false | false |
Example:
- p: The alarm is ringing
- q: There is a fire
- p ∧ q: The alarm is ringing and there is a fire
Implication
The implication operator, represented by an arrow (→), is used to denote logical implication. It evaluates to false only when the antecedent (left side) is true and the consequent (right side) is false; otherwise, it returns true. For example:
| p | q | p → q |
|---|---|---|
| true | true | true |
| true | false | false |
| false | true | true |
| false | false | true |
Biconditional
The biconditional operator, represented by a double-headed arrow (↔), evaluates to true if both operands have the same truth value (both true or both false). It returns false if the truth values are different. For example:
| p | q | p ↔ q |
|---|---|---|
| true | true | true |
| true | false | false |
| false | true | false |
| false | false | true |