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 set S is denoted as x ∈ S. If x does not belong to S, it’s denoted as x ∉ S.
The empty set or null set is a set with no elements, denoted as ∅.

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

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.

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 as P(A)), is the set of all subsets of A.
A set A is a subset of a set B (denoted as A ⊆ B), if every element of A is also an element of B.
A set A is a proper subset of a set B (denoted as A ⊂ B), is a subset that is not equal to B.
Two sets are equal (denoted as A = B), if they contain exactly the same elements.