Set Notation

What is the set A - B?

A - B := {x : x ∈ A but x not in B} (can be written A \ B)

What is the set A^c?

The compliment of the set A so all the elements not in A but in any other sets involved in the question

A ⊆ B

A is a subset of B (every element of A is also in B)

∅

The empty set (a set with no elements)

⊂

Is a proper subset of (subset but not equal)

What are the set identities?

Commutative: A u B = B u A

Associative: (A u B) u C = A u (B u C)

Distributive: (A u B)^c = A^c n B^c ; (A n B)^c = A^c u B^c

What is the set notation for R+ (positive reals)?

(0,∞)

What is the definition of a bounded set?

A set Y⊆R is bounded if there exists K ∈ R such that |x| ≤ K for all x ∈ Y. Equivalently: Y ⊆ [−K,K]

What does bounded mean?

Bounded both above and below. Some texts split the concepts: “bounded above” vs “bounded below.” The lecture uses the absolute value |x| ≤ K definition to capture both simultaneously.