Quantifiers in Mathematical Logic: Types, Notation & Examples

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  • 0:04 Quantifiers
  • 1:04 There Exists & For All
  • 2:21 Notation
  • 3:44 Other Phrases for Quantifiers
  • 5:09 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Quantifiers are special phrases in mathematics. This lesson defines quantifiers and explores the different types in mathematical logic. We also look at notation and some examples of statements.


Suppose you're talking with your friend Mary, and she is describing two clubs that she has joined. While describing the people in the first club, she says the following: 'There exists a member of Club 1, such that the member has red hair.' In describing the second club, she says the following: 'For all members in Club 2, the member has red hair.'

Based on these two statements, what can you tell me about the members' hair color in Club 1 and Club 2? Well, let's take a look at her statements, and pick them apart.

In mathematics, the phrases 'there exists' and 'for all' play a huge role in logic and logic statements. In fact, they are so important that they have a special name: quantifiers. Quantifiers are words, expressions, or phrases that indicate the number of elements that a statement pertains to. In mathematical logic, there are two quantifiers: 'there exists' and 'for all.'

There Exists & For All

The phrase 'there exists' is called an existential quantifier, which indicates that at least one element exists that satisfies a certain property. In Club 1, Mary told you that there exists a member, such that the member has red hair. This tells us that at least one member of the club has red hair, but not necessarily all of them.

The phrase 'for all' is called a universal quantifier, and it indicates that all of the elements of a given set satisfy a property. For Club 2, Mary said that 'for all members in Club 2, the member has red hair'. This tells us that all of the members of Club 2 have red hair.

A couple of mathematical logic examples of statements involving quantifiers are as follows:

  • There exists an integer x, such that 5 - x = 2
  • For all natural numbers n, 2n is an even number.

The first statement involves the existential quantifier and indicates that there is at least one integer x that satisfies the equation 5 - x = 2. The second statement involves the universal quantifier and indicates that 2n is an even number for every single natural number n.


There is a lot of explanation that goes on when writing mathematical proofs, statements, theorems, and the like. Because of this, mathematical notation is often used to shorten lengthy explanations and give your writing hand a break.

What's really neat about this is that mathematical notation is the same in every language, so mathematicians can still communicate even if they don't speak one another's language. Kind of poetic, huh?

We have symbols we use for both of our quantifiers. The symbol for the universal quantifier looks like an upside down A, and the symbol for the existential quantifier looks like a backwards E.


We can use this notation when writing statements that involve these quantifiers. For example, consider the two mathematical logic examples of statements that we gave a moment ago.

  • For all natural numbers n, 2n is an even number.
  • There exists an integer x, such that 5 - x = 2

We can rewrite these statements using our notation.

  • ∀ natural numbers n, 2n is an even number.
  • ∃ an integer x, such that 5 - x = 2

The further you go in your mathematical studies, the more notation you'll learn, and statements almost begin to look like tiny pieces of art.

Let's look at a few more examples of universal and existential quantifiers, along with their notation, to really solidify our understanding of this concept.

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