6.1 Basic set definitions

Set theory is the branch of mathematics that deals with collections of objects and the relationships between them. It provides the language used later in Venn diagrams and probability, and it is a practical tool for grouping and comparing data: suppliers who supply a given item, items stocked in a given store, or bidders who met a given requirement. This topic defines the basic terms: set, element and empty set, together with the notation used to write them.

6.1.1 Set

A set is a well-defined collection of distinct objects. Well-defined means that for any object it is possible to say clearly whether it belongs to the set or not; distinct means that no object is counted twice. Sets are named with capital letters (A, B, S), and their members are written inside curly brackets { }, separated by commas. Example: the set of suppliers prequalified to supply office furniture, S = {Supplier A, Supplier C, Supplier F}, is a set because every supplier either is or is not on the list.

Methods of describing a set

• Listing (roster) method: All the elements are written out, for example V = {a, e, i, o, u}, or the set of working days, W = {Monday, Tuesday, Wednesday, Thursday, Friday}.
• Descriptive method: The set is described in words, for example "the set of items whose stock is below the reorder level".
• Set-builder (rule) method: The set is described by a rule that its elements satisfy, for example E = {x : x is an even number, 0 < x < 10}, read "the set of all x such that x is an even number between 0 and 10", which is {2, 4, 6, 8}.

Collections that are not sets

A collection that is not well-defined is not a set. "The best suppliers in the region" is not a set, because "best" is a matter of opinion. "The suppliers who scored 70% or more in the technical evaluation" is a set, because the rule is clear.

Related terms

• Universal set (U or ξ): The set of all elements under consideration in a problem, for example all registered suppliers of an organisation. Every other set in the problem is part of it.
• Subset: A is a subset of B, written A ⊆ B, if every element of A is also an element of B. Example: the set of prequalified furniture suppliers is a subset of all registered suppliers.
• Equal sets: Two sets are equal if they have exactly the same elements, whatever the order: {1, 2, 3} = {3, 1, 2}.
• Finite and infinite sets: A finite set has a countable number of elements, such as the items in a store; an infinite set has no end, such as the set of natural numbers {1, 2, 3, ...}.
• Cardinality n(A): The number of elements in a finite set A. If A = {pens, pencils, staplers, files}, then n(A) = 4.

6.1.2 Element

An element (member) is any one of the objects that make up a set. The symbol ∈ means "is an element of" and ∉ means "is not an element of". Elements can be numbers, letters, names, items or even other sets, and each element is listed only once.

Example: let T = {cement, sand, ballast, steel bars} be the set of materials ordered for a construction project.

• cement ∈ T (cement is an element of T)
• timber ∉ T (timber is not an element of T)
• n(T) = 4 (T has four elements)

Rules about elements

• Distinct: Repeated elements are written once only; {2, 2, 3} is the same set as {2, 3}.
• Order does not matter: {cement, sand} and {sand, cement} are the same set.
• Clear membership: For every object, the rule of the set decides whether it is an element.
• Element and subset are different: cement ∈ T, but {cement} ⊆ T; an element is written alone, a subset is written in brackets.
• Counting elements: The cardinality counts distinct elements only.

Worked example

A buyer receives quotations from suppliers for five items. Let Q = {x : x is an item for which Supplier K quoted} = {paper, toner, files, pens}.

1. Is toner an element of Q? Yes, toner ∈ Q.
2. Is a stapler an element of Q? No, stapler ∉ Q, so Supplier K did not quote for staplers.
3. n(Q) = 4.
4. List the subsets of Q that contain both paper and toner: {paper, toner}, {paper, toner, files}, {paper, toner, pens}, {paper, toner, files, pens}.

6.1.3 Empty set

The empty set (null set) is the set that has no elements. It is written { } or ∅, and its cardinality is n(∅) = 0. It arises when no object satisfies the rule of a set. Example: if E is the set of bidders who submitted a tender after the closing date and every bid arrived on time, then E = ∅.

Points about the empty set

• Only one empty set: All sets with no elements are the same set, ∅.
• Subset of every set: ∅ is a subset of every set, because it has no element that is outside the other set.
• Not the same as {0}: {0} has one element, the number zero, so n({0}) = 1; ∅ has none.
• Not the same as {∅}: {∅} is a set with one element (the empty set itself).
• Use in problems: An empty set shows that a condition is never met, for example no item is both perishable and stored outdoors, or two groups have nothing in common.

Worked example

Let F = {x : x is a whole number, 5 < x < 6}. There is no whole number between 5 and 6, so F = ∅ and n(F) = 0.

Let G = {x : x is a supplier on the approved list who was debarred this year}. If no approved supplier was debarred, G = ∅.

Chapter Summary

This chapter introduced fundamental concepts of set theory starting with the definition of a set as a collection of distinct elements, including the understanding of individual elements and the special case of the empty set which contains no elements. It then explored essential operations on sets such as union, which combines all elements from two sets, intersection that identifies common elements, difference which finds elements present in one set but not the other, and symmetric difference that includes elements in either set but not in both. The chapter also covered the use of Venn diagrams as a visual tool to represent sets and their relationships clearly. Finally, practical applications of set theory were discussed, demonstrating how these concepts can be applied to solve problems involving grouping, classification, and data analysis in various contexts. Through these topics, the chapter provided a comprehensive foundation in set theory essential for further study in mathematics and related fields.