Quantifiers Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Quantifiers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Symbols specifying the scope of a predicate: \forall (for all, universal) and \exists (there exists, existential).

\forall means 'for all' (everyone). \exists means 'there exists' (at least one).

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: \forall x\,P(x) means P holds for every x; \exists x\,P(x) means at least one x makes P true. Negation swaps the quantifier.

Common stuck point: Negation: \sim(\forall x\, P(x)) = \exists x\, \sim P(x). \sim(\exists x\, P(x)) = \forall x\, \sim P(x).

Sense of Study hint: Translate the symbolic statement into plain English word by word. For negation, swap 'for all' with 'there exists' and negate the predicate.

Worked Examples

Example 1

easy
Translate into symbols and determine the truth value: (a) 'Every natural number is positive.', (b) 'There exists a real number x such that x^2 = 2.'

Solution

  1. 1
    The universal quantifier \forall means 'for all'; the existential quantifier \exists means 'there exists at least one.'
  2. 2
    Translate: (a) 'Every natural number is positive' → \forall n \in \mathbb{N},\; n > 0. (b) 'There exists a real number x such that x^2 = 2' → \exists x \in \mathbb{R},\; x^2 = 2.
  3. 3
    Truth values: (a) True under the convention \mathbb{N} = \{1,2,3,\ldots\} since all such n \ge 1 > 0. (If 0 \in \mathbb{N}, the statement is False.) (b) True: x = \sqrt{2} \in \mathbb{R} satisfies (\sqrt{2})^2 = 2.

Answer

(a)\;\forall n \in \mathbb{N},\;n>0\;(\text{True under }\mathbb{N}=\{1,2,\ldots\}),\quad (b)\;\exists x \in \mathbb{R},\;x^2=2\;(\text{True})
The universal quantifier \forall requires the predicate to hold for every element. The existential quantifier \exists requires at least one element satisfying the predicate. Truth values may depend on the domain.

Example 2

medium
Negate the statement \forall x \in \mathbb{R},\; x^2 \ge 0 and determine the truth value of both the original and its negation.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Write in words: (a) \forall x \in \mathbb{Z},\; x + 0 = x, (b) \exists x \in \mathbb{N},\; x < 5.

Example 2

medium
Determine the truth value of each and write its negation: (a) \forall x \in \mathbb{R},\; x > 0, (b) \exists x \in \mathbb{Z},\; x^2 = 3.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

logical statement