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Logic

Quantifiers

Lesson

Quantifiers say how many of something satisfy a statement. Two main flavors.

Universal ∀\forall

“For all,” “every,” “each.” To disprove: one counterexample.

Existential ∃\exists

“There exists,” “some,” “at least one.” To prove: one example.

Negation swaps them

¬(∀x P(x))≡∃x ¬P(x)\neg(\forall x \ P(x)) \equiv \exists x \ \neg P(x)
¬(∃x P(x))≡∀x ¬P(x)\neg(\exists x \ P(x)) \equiv \forall x \ \neg P(x)

How to type your answer

1 = universal, 2 = existential.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

“All squares are rectangles.”\text{``All squares are rectangles.''}

Problem 2

“Some birds can swim.”\text{``Some birds can swim.''}

Problem 3

“Every prime is greater than 1.”\text{``Every prime is greater than 1.''}

Problem 4

“There exists an even prime.”\text{``There exists an even prime.''}

Practice

Standard problems matching the lesson.

Problem 5

“All cats have whiskers.”\text{``All cats have whiskers.''}

Problem 6

“Some triangles are right triangles.”\text{``Some triangles are right triangles.''}

Problem 7

“No integer is bigger than infinity.”\text{``No integer is bigger than infinity.''}

Problem 8

“There is at least one solution.”\text{``There is at least one solution.''}

Problem 9

“Every student passed.”\text{``Every student passed.''}

Problem 10

“Some students failed.”\text{``Some students failed.''}

Problem 11

∀x∈R, x2≥0\forall x \in \mathbb{R}, \ x^2 \geq 0

Problem 12

∃x, x2=4\exists x, \ x^2 = 4

Problem 13

For all n, n+1>n\text{For all } n, \ n + 1 > n

Problem 14

Some n is even\text{Some } n \text{ is even}

Problem 15

“Each side of a square is equal.”\text{``Each side of a square is equal.''}

Problem 16

“There is an x with x>100.′′\text{``There is an } x \text{ with } x > 100.''

Problem 17

Negation of “All birds can fly” uses which quantifier?\text{Negation of ``All birds can fly'' uses which quantifier?}

Problem 18

Negation of “Some flowers are red” uses which quantifier?\text{Negation of ``Some flowers are red'' uses which quantifier?}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

∀ϵ>0 ∃δ>0…First quantifier?\forall \epsilon > 0 \ \exists \delta > 0 \ldots \text{First quantifier?}

Problem 20

Same statement. Second quantifier?\text{Same statement. Second quantifier?}

Problem 21

“There is a person who knows everyone.” First?\text{``There is a person who knows everyone.'' First?}

Problem 22

Same. Second quantifier?\text{Same. Second quantifier?}

Problem 23

¬(∀x P(x))=?\neg(\forall x \ P(x)) = ?

Problem 24

¬(∃x P(x))=?\neg(\exists x \ P(x)) = ?

Problem 25

To disprove “All swans are white” you need:\text{To disprove ``All swans are white'' you need:}

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