Logic / Equivalence Laws
Least You Need to Know: Equivalence Laws
Equivalent statements always have the same truth value. Use standard laws like **De Morgan**, **implication**, and **double negation** to rewrite expressions cleanly.
The least you need to know
- `p → q` is equivalent to `¬p ∨ q`.
- De Morgan's laws swap `and` with `or` while negating each part.
- Double negation does not change truth value.
- A biconditional can be rewritten as two implications or as matching truth values.
Key notation
p → q
¬p ∨ q
¬(p ∧ q)
¬p ∨ ¬q
¬(p ∨ q)
¬p ∧ ¬q
Tiny worked example
- Rewrite `¬(p ∧ q)`.
- By De Morgan's law, it becomes `¬p ∨ ¬q`.
- The two statements are equivalent in every row of a truth table.
Common mistakes
- Students often negate each part but forget to swap `and` and `or`.
- Students often think the converse is equivalent to the original implication.
- Students often drop parentheses and change the grouping.
How to recognize this kind of problem
- If you see `→`, consider rewriting it as `¬p ∨ q`.
- When a negation is outside parentheses, De Morgan is often the right tool.
- Equivalent rewrites preserve truth value in all cases.