Answer:
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}
