# September 28 – Set Theory

## Universal Set

## A **universal set** (usually denoted by U) is a **set** which has elements of all the related **sets**, without any repetition of elements. Say if A and B are two **sets**, such as A = {1,2,3} and B = {1,a,b,c}, then the **universal set** associated with these two **sets** is given by U = {1,2,3,a,b,c}.

## Some More Notation- Element or Not an Element

An element in a set is represented by the symbol

If something is not an element in a set we use .

**Example: Set A is {1,2,3}. We can see that 1 A, but 5 A**

**Equality = Equal Sets**

Two sets are equal if they have precisely the same members. **The equal sign (=)** is used to show equality, so we write:

A = B

### Example: Are these sets equal?

- A is {1, 2, 3}
- B is {3, 1, 2}

Yes, they are equal!

They both contain exactly the members 1, 2 and 3.

It doesn’t matter *where* each member appears, so long as it is there.

## What are Equivalent Sets? ** ↔**

In general, we can say, two sets are equivalent ** ↔ ** to each other if the number of elements in both the sets is equal. It is not necessary that they have same elements, or they are a subset of each other.

Example:

**If A = {1,2,3,4} and B = {♠♥♣♦} we can say set A is equivalent ↔ to set B because they have the same number of elements. That is because the cardinal value of both sets is 4. They symbol for equivalent sets is ↔**

Equal sets on the other hand must have the same cardinal value and the elements must be the same even if not in the same order.

## Subsets ⊂

When we define a set, if we take elements of that set, we can form what is called a **subset**.

### Example: the set {1, 2, 3, 4, 5}

A **subset** of this is {1, 2, 3}. Another subset is {3, 4} or even another is {1}, etc.

But {1, 6} is **not** a subset, since it has an element (6) which is not in the parent set.

In general:

A is a **subset **of (⊂ ) B if and only if every element of A is in B.

So let’s use this definition in some examples.

### Example: Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?

**YES! A B since all the elements of A are also in B**

**If A is not a subset of B we write it as A B.**

## Empty (or Null) Set

A null set is a set with **no elements**.

This is known as the **Empty Set** (or Null Set).There aren’t any elements in it. Not one. Zero.

**It is represented by the symbol **

**Or by { } (a set with no elements)**

**The empty set is also a subset of every set.**

## Order

In sets it **does not matter what order the elements are in**.

### Union Of Sets

The **union** of two sets A and B is the set of elements, which are in A and B . It is denoted by A ∪ B and is read ‘**A union B**’.

**Example:**

Given U = {1, 2, 3, 4, 5, 6, 7, 8, 10}

X = {1, 2, 6, 7} and Y = {1, 3, 4, 5, 8}

Find X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.

**Solution:**

X ∪ Y = {1, 2, 3, 4, 5, 6, 7, 8} ← 1 is written only once.

#### Sets: Union And Intersection

∪ is the union symbol and can be read as “or”. The union of two sets are all the elements form both sets.

∩ is the intersection symbol and can be read as “and”. The intersection of two sets are those elements that belong to both sets.

The **intersection** of two sets are those elements that belong to both sets.

The **union** of two sets are all the elements from both sets.