# Basics of Set Theory

4th May 2021

Set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts.

Fundamental set concepts

In naive set theory, a set is a collection of objects (called members or elements) that is regarded as being a single object. To indicate that an object x is a member of a set A one writes x ∊ A, while x ∉ A indicates that x is not a member of A. A set may be defined by a membership rule (formula) or by listing its members within braces. For example, the set given by the rule “prime numbers less than 10” can also be given by {2, 3, 5, 7}. In principle, any finite set can be defined by an explicit list of its members, but specifying infinite sets requires a rule or pattern to indicate membership; for example, the ellipsis in {0, 1, 2, 3, 4, 5, 6, 7, …} indicates that the list of natural numbers ℕ goes on forever. The empty (or void, or null) set, symbolized by {} or Ø, contains no elements at all. Nonetheless, it has the status of being a set.

A set A is called a subset of a set B (symbolized by A ⊆ B) if all the members of A are also members of B. For example, any set is a subset of itself, and Ø is a subset of any set. If both A ⊆ B and B ⊆ A, then A and B have exactly the same members. Part of the set concept is that in this case A = B; that is, A and B are the same set.

Representation of Sets

Sets can be represented in two ways:

• Roster Form or Tabular form
• Set Builder Form

Roster Form

In roster form, all the elements of the set are listed, separated by commas and enclosed between curly braces { }.

Example: If set represents all the leap years between the year 1995 and 2015, then it would be described using Roster form as:

A = {1996,2000,2004,2008,2012}

Also, multiplicity is ignored while representing the sets. e.g. If L represents a set that contains all the letters in the word ADDRESS, the proper Roster form representation would be

L = {A, D, R, E, S }= {S,E,D,A,R}

L≠ {A, D, D, R, E, S, S}

Set Builder Form

In set builder form, all the elements have a common property. This property is not applicable to the objects that do not belong to the set.

Example: If set S has all the elements which are even prime numbers, it is represented as:

S= { x: x is an even prime number}

where ‘x’ is a symbolic representation that is used to describe the element.

‘:’ means ‘such that’

‘{}’ means ‘the set of all’

So, S = { x:x is an even prime number } is read as ‘the set of all x such that x is an even prime number’. The roster form for this set S would be S = 2. This set contains only one element. Such sets are called singleton/unit sets.