The Set Theory
The Basics of the Set Theory
A set is a collection of elements. Elements can be anything, numbers, animals, types of numbers, colors, vehicles etc… For simplicity we would currently be referring to sets that only containing numbers. There are multiple ways to represent sets, the two of them are:
A simple list:
A range:
Both of the notations above represent the same set of numbers. The latter says that A is equal to x such that x is an element of the range 1 to 6. If the numbers in a set are within defined boundaries they can be represented in a range, if not then the set is an infinite set. An example of an infinite set is ℤ. This set includes all the integer numbers ranging from -∞ to +∞.
Example of an infinite set. The set of all positive odd integers:
Multiplying any number by two and subtracting one from it will give us an odd number. Thus 2x-1 will give us an odd number for x in the positive integer set. The Greek symbol Z is for all integers and the positive superscript is to say that it is in the positive domain. 0 < x.
Set Operations
There are a number of operation we can perform on sets just like on numbers. But in the case of sets we use different symbols. The set operators are:
The Union
The definition of union states “The action of joining together”. Similarly while dealing with sets employing the union operator would join the two sets together, but in this case multiple occurrences of identical elements in both the sets would have no effect on the number of elements in a set. The union set will only consist of unique elements from both the original sets.
A = {1, 2, 3}
B = {2, 3, 4}
A ∪ B = {1, 2, 3, 4}
Below is a Venn-diagram representation of A ∪ B:
If we see the underlying math of a union set of C and D it can be noted as:
Elements of the final union set can be in C OR can be in D.
Intersection
We can come to think about this as a crossover of two sets. The resultant set would be set of all the common elements present in both the sets and nothing more. Here the underlying math suggests that:
The elements in the final intersection set require to be in both C and D. We can take an example of two sets X and Y. If 3 is in the set X we can ask ourselves is it also in set Y, if not it would not be included in the final intersection set.
X = {1, 2, 3, 4, 5}
Y = {1, 2, 4, 7, 9}
X ∩ Y = {1, 2, 4}
X ∩ Y only consists of 1, 2 and 4 because these are the only element recurring in both X and Y.
Below is a Venn-diagram representation of X ∩ Y:
The Cartesian Product
The Cartesian Product is another set operation which would produce sets of ordered pairs which can be plotted on the Cartesian Coordinate System. The Cartesian Coordinate System is a plane which allows us to map each point by specifying the its numerical coordinates in the form (x, y). x units horizontal and y units vertical. This time the underlying math suggests that:
The above statement means that for sets C and D the Cartesian Product C × D is the set of all ordered pairs (c, d) such that c is a member of the set C and d is the member of the set D. A more helpful representation could be done using a table where:
A = {x, y , z}
B = {1, 2, 3}
A × B:
Thus the final set would be:
A × B = {(x,1), (x,2), (x, 3), (y,1), (y,2), (y,3), (z,1), (z,2), (z,3)}
All the ordered pairs of A × B can be mapped on the Cartesian plane due to being the form (x, y) and thus it has come to be called the Cartesian Product.
Set Difference
The set difference for two sets A and B would be noted as A-B. The final set when put into words is all the elements of A which are not in B.
A = {2, 3, 4}
B = {4, 5, 6}
A-B = {2, 3}
An alternative operation to set difference is the complement. The complement version of the above can be written as A ∪ B’, this says that all elements in A which are not in B.
A ∪ B’ = {2, 3}
The above are all the basics of the set theory including operation and a brief account of what are sets.
