Set Theory Introduction
Sets
and elements: A set is a collection of well-defined objects, referred
to as elements. A set may represent as Set builder form and Roster or
tabular form, for example, by a list of elements surrounded by curly
brackets and separated by commas, or using set builder notation {. .
. | . . .}, where the vertical line is an abbreviation for “such
that”. For example, {x | x is an even natural number less than 9}
and {2, 4, 6, 8} represent the same set, whose elements are precisely
2, 4 ,6 and 8.
Symbol
related to Element: The symbol ∈ is an abbreviation for “is an
element of”, and ∉ is an abbreviation for “is not an element
of”. For example, if A = {x | x is an even natural number less than
9}, then 2 ∈ A, but 5 ∉ A.
Subset
: If A and B are sets and we write A ⊆ B or B ⊇ A , then mean
every element of A is also an element of B, and say that A is a
subset of B. For example {1, 2, 3} ⊆ {1, 2, 3, 4} and {1, 2, 3, 4}
⊇ {1, 2, 3}, but {1, 2, 3, 4} ⊄ {1, 2, 3}
Equality
of sets: If A and B are sets then A = B if and only if A ⊆ B and B
⊆ A, that is, A and B have the same elements. Order and repetition
are not important. For example, {1, 2, 3, 4} = {4, 1, 3, 2} = {4, 1,
3, 1, 2, 3}.
Intersection,
union and slash: If A and B are sets then put
(a)
A ∩ B = { x | x ∈ A and x ∈ B }, called the intersection of A
and B.
(b)
A ∪ B = { x | x ∈ A or x ∈ B }, called the union of A and B.
(c)
A\B = {x | x ∈ A and x ∉ B}, called A slash B, the result
obtained after removing from A all elements from B.