Mathematical Preliminaries: Set Notation
Mathematical Preliminaries: Set Notation
Basics of Sets

A set is a collection of distinct elements that is entirely defined by its members. For example, {1, 2, 3} is a set of three distinct integers.

The notation for a set usually involves listing its elements within braces {}.

An element
x
belongs to a setS
is denoted asx ∈ S
. Ifx
does not belong toS
, it’s denoted asx ∉ S
. 
The empty set or null set is a set with no elements, denoted as ∅.
Set Operations

Union of sets
A
andB
(denoted asA ∪ B
), is a set that contains all the elements that are inA
, inB
, or in both. 
Intersection of sets
A
andB
(denoted asA ∩ B
), is a set that contains all elements thatA
andB
have in common. 
The difference of sets
A
andB
(denoted asA \ B
), is a set that contains all elements ofA
that are not inB
. 
Complement of a set
A
(denoted asA'
orA^c
), is the set of all elements that are not inA
. 
Two sets
A
andB
are said to be disjoint if their intersection is an empty set, i.e.,A ∩ B = ∅
. 
The Symmetric Difference of two sets
A
andB
(denoted asA Δ B
), is the set containing elements which are in either of the sets and not in their intersection.
Set Identities

The Commutative Law states that the order of sets does not matter in union and intersection.

The Distributive Law states that the union or intersection of sets distributes over intersection or union, respectively.

De Morgan’s Laws state that the complement of a union or intersection of sets equals the intersection or union, respectively, of their complements.
Special Sets

Universal set (denoted as
U
), is the set that includes all objects under consideration for a particular discussion or problem. 
Power set of a set
A
(denoted asP(A)
), is the set of all subsets ofA
. 
A set
A
is a subset of a setB
(denoted asA ⊆ B
), if every element ofA
is also an element ofB
. 
A set
A
is a proper subset of a setB
(denoted asA ⊂ B
), is a subset that is not equal toB
. 
Two sets are equal (denoted as
A = B
), if they contain exactly the same elements.