A set is an unordered collection of distinct objects called elements or members of the set.
Notations such as _a ∈ A _and a ∉ A are used to denote whether an element is in a set or not.
Sets are commonly denoted using uppercase letters and elements of sets with lowercase letters.
Sets can be described using the _roster method _by listing all the members of a set between braces.
Sometimes ellipses are used to describe a set when the _general pattern of the elements _is obvious.
Set builder notation is another way to describe a set by characterizing all the elements in the set by stating the property or properties they must have to be members.
Special sets include N (natural numbers), Z (integers), Q (rational numbers), R (real numbers), R+ (positive real numbers), and C (complex numbers).
Two sets are equalif they have the same elements.
The empty set (denoted by ∅ or { }) has no elements, and a set with one element is called a singleton set.
Naive set theory is the intuitive definition of a set as a collection of objects, but it can lead to paradoxes or logical inconsistencies.
VENN DIAGRAMS
Venn diagrams are a graphical representation of sets introduced by John Venn in 1881.
The universal setU, which contains all objects of interest, is represented by a rectangle.
Sets are represented inside the rectangle using circles or other geometrical figures.
Points may be used to represent the elements of a set.
Venn diagrams are commonly used to show the relationships between sets.
An example is given to draw a Venn diagram representing the set of vowels in the English alphabet.
SUBSET
A set A is a subset of a set B if every element of A is also an element of B, denoted as A ⊆ B.
B is a superset of A, denoted as B ⊇ A.
To show that A is not a subset of B, we only need to find one element in A that is not in B.
Every nonempty set has at least two subsets: the empty set and itself
proper subset is denoted as A ⊂ B, where A is a subset of B but not equal to B.
Two sets are equal_ if and only if _each is a subset of the other.
Sets may have other sets as members.
The set of letters in the English alphabet consists of 26.
The empty set has a root of 0.
set is infinite if it is not infinite.
POWER SETS
The power set of a set S is the set of all subsets of S.
To find the power set of a set, list all possible subsets of the set, including the empty set and the set itself.
The power set of the empty set is {∅}.
If a set has n elements, its power set has 2^n elements.
INTRODUCTION