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The Lumenary
Grades 6–8FoundationsPlanned

Set-builder notation, quantifiers and infinite sets

Describing sets by properties, reading and writing 'for all' and 'there exists', negating quantified statements, and meeting the first infinite sets such as the natural numbers and the even numbers.

math.found.quantifiers

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Builds on

SetsTrue, false, and, or, not

Required prerequisites, all the way down: 3

You will learn to

  • Read and write sets in set-builder notation, such as {n ∈ ℕ : n is even}.
  • Translate between everyday sentences and statements using ∀ and ∃, and negate them correctly.
  • Recognise ℕ, ℤ and the even numbers as infinite sets and explain why no finite list contains all natural numbers.

Definition

  • {x ∈ A : P(x)} is the set of those elements x of A for which P(x) is true.
  • ∀x ∈ A, P(x) means P(x) holds for every x in A; ∃x ∈ A, P(x) means P(x) holds for at least one x in A.
  • A set is finite if its elements can be listed as a₁, …, aₙ for some natural number n (n = 0 gives the empty set); otherwise it is infinite.

Theorem

  • 'Not every x has P' is equivalent to 'some x does not have P', and 'no x has P' is equivalent to 'every x fails P'.
  • The order of mixed quantifiers matters: in ℕ, ∀x ∃y (y > x) is true but ∃y ∀x (y > x) is false.
  • ℕ is infinite: given any finite list of natural numbers, one more than the largest is not on the list.

Common misconceptions

  • Swapping the order of ∀ and ∃ and expecting the meaning to stay the same.
  • Thinking a statement about all elements of the empty set must be false; it is vacuously true.
  • Treating 'infinity' as a very large natural number.

Leads to

Propositional logic and truth tablesRussell's paradox and why sets need axioms

Optional discoveries