# Sets

A (data) set is a collection of (data) elements. We can explicitly list the elements in the set or define the set by a property

The set A consists of the 6 listed elements. The set B consists of apples, pears, bananas, etc.

A data element a belongs to a set A, written a $\in$ A, provided a is a member of the set A. E.g. in the examples above, 3 $\in$ A, but 4 doesn’t belong to A (written 4 $\notin$ A).

The following are common operations on sets:

We also use the symbol Ø to represent the null set, i.e. the set containing no elements.

Sets obey a number of laws including the following (where S = the universal set containing everything under study):

An interval is the collection of values between two numbers. If a < b then we can define the following types of intervals:

The integers are a set consisting of the whole numbers = {…-3,-2 , -1, 0, 1, 2, 3, …}

### 8 Responses to Sets

1. Jan says:

Hello Charles,
A minor typo:
A∪(B∩C)=(A∪B)∩(A∪B)
should be
A∪(B∩C)=(A∪B)∩(A∪C)
In the same way:
A∩(B∪C)=⋯
Best regards
Jan

• Charles says:

Hello Jan,
Thank you very much for catching this typo. I have just changed the referenced webpage to make the necessary correction.
I really appreciate your effort to make the website better.
Charles

2. Jonathan Bechtel says:

Hi Charles,

Thank you for all this information. I’m wondering…..what does the ‘ mean in A U A’ = S??

It’s been a long time and it’s not ringing a bell. Thanks.

• Charles says:

Hi Jonathan,
A’ = the complement of A = the set of elements in the sample space S that are not in S. If S = {1,2,3,4,5} and A = {2,4} then A’ = {1,3,5}
Charles

3. Steve Gross says:

Hi Charles,

I was confused by the statement above that A = (A∩B) ∪ (A∪B′).

It seemed to me that it should be A = (A∩B) ∪ (A∩B′).

Then on the Basic Probability Concepts page I note you do state it as I thought it should be, so I believe it should be changed above.

• Charles says:

Steve,
Yes, you are correct. Thank you very much for catching this typo. I have now corrected the formula on the referenced webpage.
I appreciate your help in improving the website.
Charles

4. syed nasir shah says:

Dear Charles
I dont understand how these two are correct.(a,b)={x:a<x<b}
and [a,b]={x:a<_x<_b}
i think only one should be correct.
kindly help me.

• Charles says:

If by <_ you mean "less than or equal", then both are correct. This is because "less than" is different from "less than or equal". Charles