Logic / Quantifiers
Least You Need to Know: Quantifiers
Quantifiers tell you whether a statement is about **all** objects or about **at least one** object.
The 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
Tiny 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 this kind of problem
- 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.
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