#### SET THEORY

Set theory has its own notations and symbols that can seem unusual for many. In this tutorial, we look at some solved examples to understand how set theory works and the kind of problems it can be used to solve.

#### Definition

A set is a collection of objects.

It is usually represented in flower braces.

**For example**:

Set of natural numbers = {1,2,3,…..}

Set of whole numbers = {0,1,2,3,…..}

Each object is called an element of the set.

The set that contains all the elements of a given collection is called the universal set and is represented by the symbol ‘µ’, pronounced as ‘mu’.

For two sets A and B,

n(A ᴜ B) is the number of elements present in either of the sets A or B.

n(A ∩ B) is the number of elements present in both the sets A and B.

n(AᴜB) = n(A) + (n(B) – n(A∩B)

For three sets A, B and C,

n(AᴜBᴜC) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(C∩A) + n(A∩B∩C)

The intersection of sets is only those elements common to all sets. Let’s call our sets A, B, and C. If ‘**∩ = intersection**‘ and ‘**ᴜ = union**‘, the need-to-know formulas are:

P(A U B U C) = P(A) + P(B) + P(C) – P(A

∩B) – P(A∩C) – P(B∩C) + P(A∩B∩C)

**To find the number of people in exactly one set:**

P(A) + P(B) + P(C) – 2P(A **∩** B) – 2P(A **∩** C) – 2P(B **∩** C) + 3P(A **∩** B **∩** C)

**To find the number of people in exactly two sets:**

P(A **∩** B) + P(A **∩** C) + P(B **∩** C) – 3P(A **∩** B **∩** C)

**To find the number of people in exactly three sets:**

P(A **∩** B **∩** C)

**To find the number of people in two or more sets:**

P(A **∩** B) + P(A **∩** C) + P(B **∩** C) – 2P(A **∩** B **∩** C)

**To find the number of people in at least one set:**

P(A) + P(B) + P(C) – P(A **∩** B) – P(A **∩** C) – P(B **∩** C) + 2 P(A **∩** B **∩ **C)