Quantifiers Examples: 47 Problems with Answers

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: Quantifiers set the scope of a predicate: every case ( $\forall$ ) or at least one case ( $\exists$ ).

Common stuck point: The procedure for quantifiers is the easy part; the trap is negating 'for all' as 'for none'. Asking "Am I claiming a property for every element or for at least one element?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I claiming a property for every element or for at least one element?

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

Answer

(a) \forall n \in N, n > 0 (True under N = {1, 2, \dots}), (b) \exists x \in R, x^{2} = 2 (True)

First step

The universal quantifier

\forall

means 'for all'; the existential quantifier

\exists

means 'there exists at least one.'

Full solution

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
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 $(\sqrt{2})^2 = 2$ .

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.

Example 3

easy

Write the negation of 'Every car in the lot is red' in plain English.

Example 4

medium

Translate, then determine truth over

\mathbb{R}

: 'For every positive real

x

there is a positive real

y

with

y<x

Example 5

medium

Decide and justify over

\mathbb{R}

: '

\forall x,\ \exists y,\ y^2=x

Example 6

hard

Translate 'there is no largest integer' two ways: (a) with

\neg \exists

, (b) with

\forall \exists

Example 7

challenge

State formally that

f:\mathbb{R}\to\mathbb{R}

is not uniformly continuous on

\mathbb{R}

, by negating:

\forall \varepsilon>0,\ \exists \delta>0,\ \forall x,y,\ |x-y|<\delta \to |f(x)-f(y)|<\varepsilon

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

Example 3

easy

What does the symbol

\forall

mean?

Example 4

easy

What does the symbol

\exists

mean?

Example 5

easy

Negate the statement '

\forall x,\ P(x)

Example 6

easy

Negate the statement '

\exists x,\ P(x)

Example 7

easy

Is '

\forall x \in \mathbb{R},\ x^2 \ge 0

' true or false?

Example 8

easy

Is '

\exists x \in \mathbb{R},\ x^2 = -1

' true or false?

Example 9

easy

Translate 'every prime greater than

2

is odd' into quantifier notation (let

P(x)

= '

x

is prime',

x>2

Example 10

easy

In the statement '

\exists x,\ x+3=5

' over the integers, which

x

witnesses the existential?

Example 11

medium

Negate: '

\forall x,\ \exists y,\ x+y=0

'. Push the negation all the way in.

Example 12

medium

Are '

\forall x\, \exists y,\ y>x

' and '

\exists y\, \forall x,\ y>x

' equivalent over the reals? State which (if any) is true.

Example 13

medium

Negate 'every student passed the exam' and express it in plain English.

Example 14

medium

Translate 'there is a smallest positive integer' using quantifiers (domain: positive integers,

\le

Example 15

medium

Is the universal statement '

\forall n \in \mathbb{N},\ n^2 + n + 41

is prime' true? Justify.

Example 16

medium

Rewrite 'no real number satisfies

x^2 < 0

' with a quantifier, then give the equivalent universal form.

Example 17

medium

In '

\forall \varepsilon > 0,\ \exists \delta > 0,\ |x-a|<\delta \to |f(x)-f(a)|<\varepsilon

', does

\delta

depend on

\varepsilon

Example 18

medium

Translate 'some integer is both even and odd' and decide its truth value.

Example 19

medium

Translate 'every nonzero real has a multiplicative inverse' into quantifier notation over

\mathbb{R}

Example 20

challenge

Negate the limit definition '

\forall \varepsilon>0,\ \exists \delta>0,\ \forall x,\ (0<|x-a|<\delta \to |f(x)-L|<\varepsilon)

' fully.

Example 21

challenge

For predicate

P(x,y)

, when is '

\forall x\, \exists y\, P(x,y)

' true but '

\exists y\, \forall x\, P(x,y)

' false? Give a concrete

P

over

\mathbb{Z}

Example 22

challenge

Express 'there is exactly one

x

with

P(x)

' using

\exists

\forall

, and equality.

Example 23

easy

Translate into symbols: 'For every real number

x

x+1>x

Example 24

easy

Translate into symbols: 'There exists an integer whose square is

9

Example 25

easy

True or false over

\mathbb{R}

\exists x,\ x^2=2

Example 26

easy

True or false over

\mathbb{Z}

\forall n,\ 2n

is even.

Example 27

easy

Give a witness that makes '

\exists x \in \mathbb{Z},\ x^2-4=0

' true.

Example 28

medium

Negate and simplify: '

\forall x \in \mathbb{R},\ (x>0 \to x^2>0)

Example 29

medium

Are '

\forall x \exists y,\ y=x+1

' and '

\exists y \forall x,\ y=x+1

' equivalent over

\mathbb{Z}

? Briefly justify.

Example 30

medium

Translate: 'Every nonempty set of positive integers has a least element' using

\forall,\exists,\in

Example 31

medium

Disprove '

\forall n \in \mathbb{N},\ n^2 \ge 2n

' by giving a counterexample or show it is true.

Example 32

medium

Translate: 'No prime greater than

2

is even' using a universal quantifier.

Example 33

medium

Negate, pushing

\neg

inward: '

\exists x \forall y,\ x \le y

' over

\mathbb{Z}

Example 34

medium

Translate: 'The function

f

is surjective from

A

B

' using quantifiers.

Example 35

medium

Translate: '

f

is injective on

A

' using quantifiers.

Example 36

medium

Translate 'every integer is even or odd' and write its negation.

Example 37

hard

Negate the convergence statement '

\forall \varepsilon>0,\ \exists N \in \mathbb{N},\ \forall n \ge N,\ |a_n - L|<\varepsilon

Example 38

hard

Over

\mathbb{R}

, decide and justify: '

\exists x,\ \forall y,\ x+y>y

Example 39

hard

Let

P(x,y)

be '

x

divides

y

' on

\mathbb{Z}^+

. Decide truth: (a)

\forall y,\ \exists x,\ P(x,y)

; (b)

\exists x,\ \forall y,\ P(x,y)

Example 40

challenge

Translate uniqueness '

\exists! x,\ P(x)

' using only

\exists,\forall,\to,=

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Related Concepts

Logical Statement Proof Techniques

Background Knowledge

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

logical statement