Logic is the language mathematicians use to decide whether statements are true or false. A proposition is a statement that has a definite truth value, such as P or Q. Truth tables organize every possible combination of truth values so you can test compound statements carefully.
This matters because logic supports proofs, computer circuits, programming conditions, and clear mathematical reasoning.
Logical connectives build new statements from simpler ones. NOT reverses truth, AND requires both parts to be true, OR requires at least one part to be true, and conditional statements track whether one statement guarantees another. A truth table works like a logic machine: list all possible inputs, apply each connective step by step, and compare output columns.
If two statement forms always have the same final truth values, they are logically equivalent.
Key Facts
- NOT: ¬P is true when P is false, and false when P is true.
- AND: P ∧ Q is true only when both P and Q are true.
- OR: P ∨ Q is true when P is true, Q is true, or both are true.
- Conditional: P → Q is false only when P is true and Q is false.
- Biconditional: P ↔ Q is true when P and Q have the same truth value.
- Logical equivalence: A ≡ B means A and B have identical truth table columns.
Vocabulary
- Proposition
- A proposition is a statement that is either true or false, but not both.
- Truth value
- A truth value is the label true or false assigned to a proposition or compound statement.
- Logical connective
- A logical connective is a symbol such as ∧, ∨, ¬, →, or ↔ that combines or modifies propositions.
- Truth table
- A truth table lists all possible truth value combinations for propositions and shows the resulting truth values of compound statements.
- Logical equivalence
- Logical equivalence means two statements have the same truth value in every possible case.
Common Mistakes to Avoid
- Treating P ∨ Q as exclusive or is wrong because in standard logic OR is true when both P and Q are true.
- Marking P → Q false whenever Q is false is wrong because a conditional is false only in the case P is true and Q is false.
- Skipping intermediate columns in a truth table is risky because complex statements like ¬(P ∧ Q) require one operation at a time.
- Confusing ¬(P ∧ Q) with ¬P ∧ ¬Q is wrong because De Morgan's law says ¬(P ∧ Q) ≡ ¬P ∨ ¬Q.
Practice Questions
- 1 Make a truth table for P ∧ ¬Q with columns P, Q, ¬Q, and P ∧ ¬Q. How many rows have a true final result?
- 2 Make a truth table for (P → Q) ↔ (¬Q → ¬P). Is the final column true in all 4 rows?
- 3 Explain why P → Q does not mean the same thing as Q → P. Give one truth value assignment where they differ.