Logic / Quantifiers
Least You Need to Know: Quantifiers
Quantifiers tell you whether a statement is about all objects or about at least one object.
Least you need to know
- ∀ means 'for all'.
- ∃ means 'there exists'.
- Negating ∀ turns it into ∃.
- Negating ∃ turns it into ∀.
- Negate the predicate exactly, not approximately.
Key notation
- ∀ — for all
- ∃ — there exists
- ¬ — not
- → — implies
Worked example
- Statement: For every integer n, n^2 >= n.\n- Negation: There exists an integer n such that n^2 < n.\n- Notice that the quantifier changed and the predicate was negated exactly.
Common mistakes
- Students often change the predicate but forget to change the quantifier.
- Students often negate >= as <= instead of <.
- Students often treat 'for every' as if it means 'for many'.
How to recognize it
- Look for words like every, each, all, some, or there exists.
- If you are asked to negate a statement, handle the quantifier first.
- If nested quantifiers appear, negate one layer at a time.
Next recommended lesson
Continue through this topic with Least You Need to Know: Truth Tables and Equivalence.
Least You Need to Know: Truth Tables and EquivalenceRelated lessons
Keep going with nearby lessons in the same topic.