# Set Notation,Math Sets and Subsets

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 Mathwords website.

## 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  JK.

### 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  Ac.

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

Set J  minus  Set Kor  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 }

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.

The set  MN  =  { 2 , 7 , 9 , 11 , 14 , 65 , 71 }.

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