Practice Quantifiers in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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).

Showing a random 20 of 50 problems.

Example 1

easy
Fill in: The negation of ∃x, P(x)\exists x,\ P(x) is ____.

Example 2

easy
True or false over Z\mathbb{Z}: ∀n, 2n\forall n,\ 2n is even.

Example 3

medium
Fill in: ¬(∀x, P(x)∧Q(x))\neg(\forall x,\ P(x) \land Q(x)) is equivalent to ∃x, □\exists x,\ \square. Fill the □\square.

Example 4

challenge
Express 'there is exactly one xx with P(x)P(x)' using ∃\exists, ∀\forall, and equality.

Example 5

easy
Is '∀x∈R, x2≥0\forall x \in \mathbb{R},\ x^2 \ge 0' true or false?

Example 6

easy
Translate into symbols and determine the truth value: (a) 'Every natural number is positive.', (b) 'There exists a real number xx such that x2=2x^2 = 2.'

Example 7

medium
Translate: 'The function ff is surjective from AA to BB' using quantifiers.

Example 8

medium
Disprove '∀n∈N, n2≥2n\forall n \in \mathbb{N},\ n^2 \ge 2n' by giving a counterexample or show it is true.

Example 9

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

Example 10

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

Example 11

medium
Rewrite 'no real number satisfies x2<0x^2 < 0' with a quantifier, then give the equivalent universal form.

Example 12

challenge
Translate uniqueness '∃!x, P(x)\exists! x,\ P(x)' using only ∃,∀,→,=\exists,\forall,\to,=.

Example 13

medium
Translate, then determine truth over R\mathbb{R}: 'For every positive real xx there is a positive real yy with y<xy<x.'

Example 14

hard
Over R\mathbb{R}, decide and justify: '∃x, ∀y, x+y>y\exists x,\ \forall y,\ x+y>y.'

Example 15

easy
Translate 'every prime greater than 22 is odd' into quantifier notation (let P(x)P(x) = 'xx is prime', x>2x>2).

Example 16

medium
Determine the truth value of each and write its negation: (a) ∀x∈R,  x>0\forall x \in \mathbb{R},\; x > 0, (b) ∃x∈Z,  x2=3\exists x \in \mathbb{Z},\; x^2 = 3.

Example 17

easy
What does the symbol ∀\forall mean?

Example 18

hard
Translate 'there is no largest integer' two ways: (a) with ¬∃\neg \exists, (b) with ∀∃\forall \exists.

Example 19

easy
Fill in: The negation of ∀x, P(x)\forall x,\ P(x) is ____.

Example 20

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