Contents

**Definition**

A **set **is a
collection of anything: numbers, letters, objects, etc.

The members, or elements, of the sets are written between braces like this: {1, 2, 3, 4, 5}.

The elements of this set are simply the numbers 1, 2, 3, 4, and 5. Another example of a set is {apples, peaches, pears}.

**Two
sets are equal**

Two sets are equal if they have the same elements. The order in which the elements of the set are listed does not matter. Thus {1, 2, 3, 4, 5} = {5, 4, 3, 2, 1}. We can use one letter to stand for a whole set; for example, A = {1, 2, 3, 4, 5}.

**The
union of two sets**

– The union of two sets is a set, which **contains** *all elements of two sets*.

– The union of sets A and B is written **A∪****B**.

**Example:**

If A = {1, 2, 3, 4} and B = {2, 4, 6}, find A∪B.

All the elements in either A or B or both are 1, 2, 3, 4, and 6.

Therefore **A∪B = {1, 2, 3, 4, 6}.**

**The
intersection of two sets**

– The intersection of two sets is a set, which **contains** the elements in both two sets

– The intersection of sets A and B is written A∩B.

**Example:**

If A = {1, 2, 3, 4} and B = {2, 4, 6}, find A∩B.

The elements in both A and B are 2 and 4.

Therefore **A**∩**B = {2, 4**}.

*If two sets have no elements in common, then their intersection is the null or empty set written as ∅ or { }.*

**Example:**

If A= {1, 3, 5, 7} and B = {2, 4, 6, 8}, find AB.

There is no elements in common in both A and B.

Therefore **A**∩**B = {**∅**}**.

**To
perform several union and intersection operations**

First operate on sets within parentheses.

**Example:**

If *A *= {1, 2, 3} and *B *= {2, 3, 4, 5, 6} and *C *= {1, 4, 6} find *A***∪**(*B*∩*C*).

First we find *B*∩*C *by listing all the elements in both *B *and *C*. *B*∩*C *= {4, 6}.

Then *A***∪**(*B*∩*C*) is just the set of all members in at least one of the sets *A *and {4, 6}.

Therefore, *A***∪**(*B*∩*C*) = {1, 2, 3, 4, 6}.

**A
subset**

– A subset of a set is a set, all of whose members are in the original set. Thus, {1, 2, 3} is a subset of the set {1, 2, 3, 4, 5}.

– Note that the null set is a subset of every set, and also that every set is a subset of itself.

In general, a set with ** n** elements has

**2**.

*subsets*^{n}**Example:**

How many subsets does {*x*, *y*, *z*} have?

This set has 3 elements and therefore **2 ^{3}**, or

**8 subsets.**

They are: {x}, {y}, {z}, {x,y}, {y,z}, {x,z}, {x,y,z}, {∅}