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Existential Quantifiers: "Some"

A free Logic lesson from the “Quantifiers and Predicates” unit, with a worked example and practice problems including step-by-step solutions.

An existential statement claims at least one object in the domain has a property. One example is enough to make it true. Learning objective: Translate and test some statements. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: 'All integers are rational' is universal. Example 2: 'Some rectangles are squares' is existential because it claims at least one example. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.

What you'll learn

Why it matters: All and some language appears in data claims, geometry theorems, function domains, and proof statements.

Worked example

Problem. Example case A (Existential Quantifiers: "Some"): Worked example: Which phrase signals an existential statement?

  1. Existential statements claim an example exists.
  2. Some, at least one, and there exists are existential signals.
  3. Every and all are universal.

Answer: at least one

Practice problems

1. Practice case A (Existential Quantifiers: "Some"): Practice: Which phrase signals an existential statement?

Choices: at least one · every · all · no exceptions

Show solution
  1. Existential statements claim an example exists.
  2. Some, at least one, and there exists are existential signals.
  3. Every and all are universal.

Answer: at least one

2. Practice case B (Existential Quantifiers: "Some"): Practice: Which statement is existential?

Choices: Some rectangles are squares. · All rectangles have four sides. · Every square is a rectangle. · No triangles are circles.

Show solution
  1. Existential statements use some, at least one, or there exists.
  2. They claim one or more examples exist.
  3. The other three are universal or universal-negative.

Answer: Some rectangles are squares.

3. Practice case C (Existential Quantifiers: "Some"): Practice: Translate "some integer x satisfies x² = 9" into quantifier language.

Choices: There exists x (∃x) such that x² = 9. · For all x (∀x), x² = 9. · For no x, x² = 9. · Every x satisfies x² = 9.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Existential Quantifiers: "Some"): Practice: Translate "some integer x satisfies x² = 9" into quantifier language.
  2. For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
  3. Some means at least one example exists.
  4. The symbol for there exists is ∃.
  5. So the statement uses ∃x.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: There exists x (∃x) such that x² = 9.

4. Practice case D (Existential Quantifiers: "Some"): Practice: What does "∃x in the integers, x² = x" say in plain English?

Choices: At least one integer equals its own square. · Every integer equals its own square. · No integer equals its own square. · All integers equal their own square.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case D (Existential Quantifiers: "Some"): Practice: What does "∃x in the integers, x² = x" say in plain English?
  2. For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
  3. ∃ means there exists.
  4. It claims at least one example.
  5. So it reads as a some/at-least-one statement.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: At least one integer equals its own square.

5. Practice case E (Existential Quantifiers: "Some"): Practice: What would prove "Some odd numbers are prime"?

Choices: Finding one odd number that is prime · Showing every odd number is prime · Showing no odd number is prime · Restating the claim in symbols

Show solution
  1. Some means at least one.
  2. A single confirming example proves an existential claim.
  3. You do not need to check the whole domain.

Answer: Finding one odd number that is prime

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