Set-builder notation and matching infinite sets
Know ℕ, ℤ, ℚ and ℝ, list the members of a set from its condition, write the condition for a set, and see why the evens and the naturals are "the same size".
- France: grade 10
- Iran: grade 9
- Ukraine: grade 8
Guide
Number sets: ℕ natural numbers (1, 2, 3, …), ℤ integers, ℚ rational numbers and ℝ real numbers. ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ: each lies in the next.
We can write a set by its condition: A = {x ∈ ℤ | −2 < x ≤ 3} means "A is the set of all integers x greater than -2 and at most 3". Its members are {−1, 0, 1, 2, 3}.
Watch the sign: < leaves the number out and ≤ keeps it. And ℝ cannot be listed by members; we write it with a condition.
We compare two infinite sets by pairing their members one to one. If every member can be paired and none is left over, the sets are "the same size": n pairs with 2n.