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.
Least you need to know
p → qis equivalent to¬p ∨ q.- De Morgan's laws swap
andwithorwhile 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
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
andandor. - Students often think the converse is equivalent to the original implication.
- Students often drop parentheses and change the grouping.
How to recognize it
- 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.
Next recommended lesson
Continue through this topic with Least You Need to Know: Nested Quantifiers.
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