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Glacius
FoundationsConcept reference

Quantifiers

A universal claim requires a property for every member of a domain, whereas an existence claim requires at least one qualifying member.

2 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. A quantifier tells you how broadly a claim applies
  3. Suppose file A opens and B and C have not been checked
  4. Check your understanding
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections
Two schematic domains of three members. The universal condition highlights every member; the existence condition highlights one witness, without ruling out others.Two schematic domains of three members. The universal condition highlights every member; the existence condition highlights one witness, without ruling out others.
Figure 1Two schematic domains of three members. The universal condition highlights every member; the existence condition highlights one witness, without ruling out others.
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01A quantifier tells you how broadly a claim applies#

A quantifier tells you how broadly a claim applies. Compare two claims about a nonempty set of three files: “Every file opens” and “At least one file opens.” They ask for different evidence.

A universal claim requires the property for every member of its domain. If DD is a set of three files, “Every file in DD opens” covers those files, not every file anywhere.

The universal symbol reads “for every”.

∀x∈D: x opens\forall x\in D:\ x\text{ opens}

For a set DD of three files, “At least one file in DD opens” is an existence claim. It needs a member with the property. “Some” means at least one, which may include all.

The existence symbol reads “there exists”.

∃x∈D: x opens\exists x\in D:\ x\text{ opens}
Check your reasoning

Which claim concerns every card?

  1. AEvery card is blue.
  2. BA blue card exists.
  3. CAt least one card is blue.
Show answer and explanation
Every card is blue.

Every covers the whole set.

02Suppose file A opens and B and C have not been checked#

Suppose file A opens and B and C have not been checked. This establishes that a file opens. It leaves “Every file opens” unresolved. Distinguishing the claim’s scope comes before judging it true or false.

Check your reasoning

Six sensors: one responds, five untested. Which claim follows?

  1. AAll respond.
  2. BAt least one responds.
  3. CExactly one responds.
Show answer and explanation
At least one responds.

One responds. Others are untested.

03Check your understanding#

Check your reasoning

All three cards are blue. “At least one is blue” requires:

  1. AExactly one blue card.
  2. BOne or more blue cards.
  3. COne non-blue card.
Show answer and explanation
One or more blue cards.

One or more allows all three cards.

Key takeaway

Name the domain and read the quantifier. Every requires the whole domain; exists or some requires at least one and allows more.

  • Distinguish a universal claim from an existence claim

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Quantifiers.” Math behind ML. /learn/f-quantifiers