# 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 ∅.

## Set Operations

• 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.

## 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 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.