Things You Should Know

I will assume students are familiar with all of these terms and symbols. Please don’t hesitate to ask about anything that seems unfamiliar. In the following, A,B,X, Y are sets. I is an index set, {Aα : α ∈ I } and {Bα : α ∈ I } are families of sets indexed by I, and C is a collection of sets. • Empty set: ∅, the set with no elements. • Subset: A ⊆ B means “x ∈ A⇒ x ∈ B” • Power set: P(X) := {A : A ⊆ X } • Union: A ∪B := {x : x ∈ A or x ∈ B } • Intersection: A ∩B := {x : x ∈ A and x ∈ B } • Complement: If A ⊆ X, then X \A := {x : x ∈ X and x / ∈ A } • Indexed union: ⋃ α∈I Aα := {x : ∃α ∈ I, x ∈ Aα } • Non-indexed union: ⋃ C := ⋃ X∈C X. • Indexed intersection: ⋂ α∈I Aα := {x : ∀α ∈ I, x ∈ Aα } • Non-indexed intersection: ⋂ C := ⋂ X∈C X. • Cartesian product of two sets: X × Y := { (x, y) : x ∈ X, y ∈ Y }