Two sets are equal if they contain the

same elements. I. e., sets A and B are equal if

∀x[x ∈ A ↔ x ∈ B].

Notation: A = B.

Recall: Sets are unordered and we do not distinguish

between repeated elements. So:

{1, 1, 1} = {1}, and {a, b, c} = {b, a, c}.

Definition: Two sets are equal if they contain the

same elements. I. e., sets A and B are equal if

∀x[x ∈ A ↔ x ∈ B].

Notation: A = B.

Recall: Sets are unordered and we do not distinguish

between repeated elements. So:

{1, 1, 1} = {1}, and {a, b, c} = {b, a, c}.

Definition: A set A is a subset of set B, denoted

A ⊆ B, if every element x of A is also an element of B.

That is, A ⊆ B if ∀x (x ∈ A → x ∈ B).

Example: Z ⊆ R.

{1, 2} ⊆ {1, 2, 3, 4}

Notation: If set A is not a subset of B, we write A 6⊆ B.

Example: {1, 2} 6⊆ {1, 3}