Logical operators are core tools in propositional logic. They let you combine simpler statements into more complex expressions and then reason about their overall truth.
In this module, you will review the most common logical operators and how they behave, so you can apply them in discrete mathematics, proofs and computer science.
"Once you understand logical operators, you can start thinking like both a mathematician and a computer."
🔎 What you will learn
- 🔧 How AND, OR and NOT combine or negate simple propositions
- ➡️ How IMPLIES captures conditional statements like "if p then q"
- 🔁 How IFF represents logical equivalence between two statements
- ❌ How XOR captures the idea of "either p or q, but not both"
🔧 AND ( ∧ )
The "AND" operator is true only when both propositions are true. If p and q are both true, then p ∧ q is true. In all other cases, it is false.
🚪 OR ( v )
The "OR" operator is true if at least one operand is true. p v q is true when p is true, q is true, or both are true.
🚫 NOT ( ¬ )
The "NOT" operator reverses the truth value. ¬p is true if p is false, and false if p is true.
➡️ Implies ( → )
The "implies" operator represents a conditional statement. p → q is false only when p is true and q is false. In all other cases, it is true.
🔁 If and only if ( ↔ )
This operator is true when both operands have the same truth value. p ↔ q is true when both are true or both are false. It represents logical equivalence.
❌ Exclusive OR ( ⊕ )
The "XOR" operator is true when p and q have different truth values. p ⊕ q is true when one is true and the other is false.
✅ Key takeaway
Logical operators are the foundation of proofs, circuits, programming conditions and formal reasoning. Mastering their behaviour makes it easier to analyse and build complex systems.
