Discrete Structures Connectives (Defn Focus)

Includes Definitions, Representation (Symbols), and Truth Condition.

Negation

The NOT operation (¬): Reverses the truth of a single proposition.

Truth Condition: ¬P is True when P is False.

Conjunction

The AND operation (∧): Both components must be True for the output to be True.

Truth Condition: True exactly when P is True and Q is True.

Disjunction

The OR operation (∨): At least one component is True, then the output is True.

Truth Condition: True when at least P or Q is True.

Implication (aka Conditional)

A guarantee. If the first statement holds, the second must not fail.

Representation: P → Q - “P implies Q” or “if P then Q”
Truth Condition: False only when P is True and Q is False.

Biconditional

A two-way guarantee. True when both parts are True.

Representation: P ↔ Q - “P … if and only if Q …”
Truth Condition: True when both are True or both are False.

Exclusive-or (XOR)

Exactly one component is True. Either-or choices cannot both hold. 

Representation: P ⊕ Q or P xor Q
Truth Condition: True when P and Q differ.

NAND

The negation of conjunction (↑)

Truth Condition: True except when P and Q are True.

NOR

The negation of disjunction (↓)

Truth Condition: True exactly when P and Q are False.