Universal Quantifiers: "All"
A free Logic lesson from the “Quantifiers and Predicates” unit, with a worked example and practice problems including step-by-step solutions.
A universal statement claims every object in the domain has a property. One counterexample is enough to make it false. Learning objective: Translate and test all 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
- Translate and test all statements
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Universal Quantifiers: "All"): Worked example: Which phrase signals a universal statement?
- Universal statements cover the whole domain.
- Words like all and every are universal signals.
- Some and at least one are existential.
Answer: all
Lesson preview: this page shows 5 practice items. The interactive activity contains 20 practice items and 10 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Practice case A (Universal Quantifiers: "All"): Practice: Which phrase signals a universal statement?
Choices: all · some · at least one · there exists
Show solution
- Universal statements cover the whole domain.
- Words like all and every are universal signals.
- Some and at least one are existential.
Answer: all
2. Practice case B (Universal Quantifiers: "All"): Practice: Which statement is universal?
Choices: All rectangles have four sides. · Some rectangles are squares. · At least one rectangle is blue. · There exists a red square.
Show solution
- Universal statements use all or every.
- They make a claim about every object in the domain.
- The other three claim only that an example exists.
Answer: All rectangles have four sides.
3. Practice case C (Universal Quantifiers: "All"): Practice: Translate "every integer x satisfies x + 0 = x" into quantifier language.
Choices: For all x (∀x), x + 0 = x. · There exists x (∃x) such that x + 0 = x. · For no x, x + 0 = x. · Not all x satisfy x + 0 = x.
Show solution
- Every means universal.
- The symbol for for all is ∀.
- So the statement uses ∀x.
Answer: For all x (∀x), x + 0 = x.
4. Practice case D (Universal Quantifiers: "All"): Practice: What does "∀x in the integers, x + 1 > x" say in plain English?
Choices: Every integer is less than the next integer. · Some integer is less than the next integer. · No integer is less than the next integer. · Exactly one integer is less than the next.
Show solution
- ∀ means for all.
- The claim applies to every integer.
- So it reads as an every/all statement.
Answer: Every integer is less than the next integer.
5. Practice case E (Universal Quantifiers: "All"): Practice: Which value is a counterexample to "All odd numbers are prime"?
Choices: 9 · 3 · 5 · 7
Show solution
- A counterexample is one odd number that is not prime.
- 9 is odd but not prime, since 9 = 3 × 3.
- One counterexample is enough to disprove an all claim.
Answer: 9
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