Post 4: Set / Set Notation / Set Operation

Set:

A set is a collection of distinct objects. The objects in a set are called elements of the set. Note that order of appearance of elements in a set is immaterial

There are 2 alternative ways of writing a set: by enumeration and by description.

Enumeration: If we let S represent the set of three numbers 2, 3, and 4, we can write, by enumeration of the elements,
                                 S = {2, 3, 4}

Description: But if we let I denote the set of all positive integers, enumeration is not feasible, and we may instead simply describe the elements and write

                                 I = {x | x is a positive integer}

Membership in a set is indicated by the symbol - ∈.

T is a subset of S if and only if x ∈ T implies x ∈ S. Using the set inclusion symbol ⊂ (is contained in) and ⊃ (includes), we may then write: T ⊂ S or S ⊃ T.

Any subset that doesn't contain all the elements of S is called a proper subset of S. The largest possible subset of S is the set S itself. The smallest possible subset of S is a set that contains no element at all. Such a set is called null set, or empty set, denoted by the letter ∅ or {}.

Counting all the subsets of S, including the two limiting elements, S and ∅, we find a total of  subsets of set S.

Two sets may have no elements in common at all. In that case, the two sets are said to be disjoint.

Set Operation:

Intersection: A ∩ B = {x | x ∈ A and x ∈ B}
Union: A ∪ B = {x | x ∈ A or x ∈ B}
Complement: Ã = {x | x ∈ U and x ∉ A}

Commutative Law: A ∪ B = B ∪ A     A ∩ B = B ∩ A
Associative Law:
                                          A ∪ (B ∪ C) = (A ∪ B) ∪ C
                                          A ∩ (B ∩ C) = A ∩ (B ∩ C)
Distributive Law:

                                          A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) 
                                          A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)