Logic / Truth Tables
Least You Need to Know: Truth Tables and Equivalence
Truth tables let you compare two statements row by row. Two statements are logically equivalent when they match on every row.
Least you need to know
- A truth table checks every possible truth assignment.
- If two columns match in every row, the statements are equivalent.
- An implication is false only when the premise is true and the conclusion is false.
- A tautology is always true; a contradiction is always false.
- Equivalence is about matching truth values, not just sounding similar.
Key notation
- ¬p — not p
- p ∧ q — p and q
- p ∨ q — p or q
- p → q — if p then q
- p ↔ q — p if and only if q
Worked example
- Compare
p → qwith¬p ∨ q. - In every row of the truth table, these columns match.
- So
p → qand¬p ∨ qare logically equivalent.
Common mistakes
- Students often think one matching row is enough for equivalence.
- Students often forget the special false row for implication.
- Students often confuse
↔with→.
How to recognize it
- The prompt asks whether two statements are equivalent.
- You are asked whether a statement is a tautology or contradiction.
- The problem mentions all possible truth values or rows.
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