**On this Page**:1. Set Notation, Further

2. Finite, Infinite Sets

3. Subsets, Set Compliment

4. Set Intersection & Union

This page will look to introduce the concept of Math sets and subsets, and the notation that is involved.

## Set Notation

In Math, a set is a collection of objects or elements.

In set notation, the common notation for a set, is a collection of elements inside curly brackets, separated by commas.

For example, **{ 1 , 2 , 3 , 4 }**.

The letters from A to E can also be a set.

**{ A , B , C , D , E }**

The set order doesn’t matter, **{ B , D , C, A , E }** is still the same set.

## Sets and Set Notation, Further

We could have a situation where we have 2 sets, A and B.

A =

**{ 10 , 11 , 12 , 13 , 14 }**

B =

**{ 15 , 16 , 17 , 18 , 19 }**

The number

**11**is a member of set A. This would be denoted by

**11 ∈**A.

**∈**= “is a member/element of”.

As can be seen, it’s also the case that

**17 ∈**B.

But, the number

**12**is

__NOT__an element of the set B,

likewise the number

**16**is

__NOT__an element of the set A.

These cases can be denoted by

**12 ∉**B and

**16 ∉**B.

**∉**= “is NOT a member/element of”.

There is also the case where a set can have no elements at all in it.

When this happens, such a set is known as the empty set { }, which is denoted by the symbol **∅**.

## Finite and Infinite Sets

It’s the case in Math that sets don’t have to only be finite groups, they can also be infinite.

For instance, we could have a set that is all whole numbers larger than **10**.

**{ 11 , 12, 13 , …. }**

There is set notation that can make presenting sets a little bit easier and shorter, particularly for cases where sets are infinite.

A set for all the numbers greater in size than

**7**can be denoted as:

**{**.

*y*|*y*> 7 }Which means that we have a set in which the letter * y* can be any number, but it must be greater than

**7**.

A handy page with some good examples set notation is available to view at the

__website.__

*Mathwords*## Math Sets and Subsets,

Set Compliment

### Sets and Subsets:

With Math sets and subsets, a set J can be a ‘subset’ of another set K.

If it happens to be the case that all of the elements that are in set **J**, are elements that are also in set **K**.

We can look at 2 sets.

J =

**{ 3, 12 , 24 }**K =

**{ 3 , 8 , 12 , 24 , 35 }**

In this situation, set J is a subset of set K.

The notation for this is J ⊂ K.

### Set Compliment:

With sets in Math, the ‘complement’ of a set, is the elements that are __NOT__ part of that set.

If a set is labelled as A, the notation for the compliment would be A^{c}.

But also, when you have 2 separate sets, there can be another set that is either:

Set J

*minus*Set K,

__or__Set J

*minus*Set K.

The notation for which is a backslash between the set labels.

J\K = Elements in J, but not in K.

K\J = Elements in K, but not in J.

Thus for the sets J and K here, the set K\J is:

**{ 8 , 35 }**.

As

**8**and

**35**are in set K, but

__NOT__in set J.

## Math Sets and Subsets,

Intersection and Union

If we had two sets denoted as M and N.

The ‘intersection’ of both sets, is the set of elements in both set M and set N.

The notation for which is M ∩ N.

If we have 2 sets:

M =

**{ 11 , 2 , 7 , 65 , 14 }**

N =

**{ 9 , 2 , 71 , 65 , 14 }**

Then the set M ∩ N =

**{ 2 , 14 , 65 }**.

The union of two sets is the set of elements that are in either set M __or__ in set N.

The notation for which is M∪N.

**{ 2 , 7 , 9 , 11 , 14 , 65 , 71 }**.

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