Mathematical Analysis I, 2004a

Chapter 1
Set Theory
§§1–3. Sets and Operations on Sets. Quantifiers
A set is a collection of objects of any specified kind. Sets are usually denoted
by capitals. The objects belonging to a set are called its elements or members.
We write x ∈ A if x is a member of A, andx ∉ A if it is not.
A = {a, b, c, ...} means that A consists of the elements a, b, c, .... In
particular, A = {a, b} consists of a and b; A = {p} consists of p alone. The
empty or void set, ∅, hasno elements. Equality (=) means logical identity.
If all members of A are also in B, wecallA a subset of B (and B a superset
of A), and write A ⊆ B or B ⊇ A. It is an axiom that the sets A and B are
equal (A = B) if they have the same members, i.e.,
A ⊆ B and B ⊆ A.
If, however, A ⊆ B but B ⊈ A (i.e., B has some elements not in A), we call A
a proper subset of B and write A ⊂ B or B ⊃ A. “⊆” is called the inclusion
relation.
Set equality is not affected by the order in which elements appear. Thus
{a, b} = {b, a}. Notsoforordered pairs (a, b). 1 For such pairs,
(a, b) =(x, y) iff 2 a = x and b = y,
but not if a = y and b = x.
Similarly, for ordered n-tuples,
(a 1 ,a 2 , ..., a n )=(x 1 ,x 2 , ..., x n ) iff a k = x k , k =1, 2, ..., n.
We write {x | P (x)} for “the set of all x satisfying the condition P (x).”
Similarly, {(x, y) | P (x, y)} is the set of all ordered pairs for which P (x, y)
holds; {x ∈ A | P (x)} is the set of those x in A for which P (x) istrue.
1 See Problem 6 for a definition.
2 Short for if and only if ; also written ⇐⇒.