8. Set Theory & Data Analysis
A collection of non-repeating (unique) elements is called a set.
e.g. The set of letters in "November" is {$N,O,V,E,M,B,R$}
A set X that is completely contained in another set Y is called a proper subset.
X is a "simple" subset of itself.

The set of common elements between two sets X and Y is called their intersection set
The set of all elements in X, in Y, and in their intersection, is called their set union
The set of all elements in X but not in Y is denoted by $X-Y$ and is called a set difference
Please note that X-Y may not equal Y-X. e.g. $X$={$1,2,4,5$}, $Y$=[$2,3,5,6$}. $X-Y$={$1,4$}, while $Y-X$={$3,6$}
The number of elements of a set is called its cardinality.
The set of all elements under consideration is called a sample space

We can draw sets and subsets in a sample space and draw interesting inferences about our data. This diagram is called a Venn Diagram

A Venn diagram makes relationships between sets clear. For example:
If 150 cars are registered today at a DMV of which 30 are Toyotas, and 30 are red, and 5 are both, then how many are neither?
Total sample space=150 cars, X(Red)=30, Y(Toyota)=30, intersection(X,Y)=5
So, Red non-Toyotas=$30-5=25$, Non-Red Toyotas=$30-5=25$.
Which leaves a total of $150-(25+25+5)=95$ cars that are non-red, non-Toyotas of the 150 cars registered today.


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