Learning Math through Set Theory

In grade school, we’re taught that math is about numbers. When we get to college (the ones of us who are still interested in math), we’re taught that mathematics is about sets, operations on sets and properties of those sets.

Understanding Set Theory is fundamental to understanding advanced mathematics. Iv wrote these scripts so that users could begin to play with the different set operations that are taught in a basic set theory course. Here, the sets are limited to positive integers and we’re only looking at a few operations, in particular the union, intersection, difference, symmetric difference, and cross product of two sets. I will explain what each of these is below.

The union of the sets S1 and S2 is the set S1 [union] S2, which contains the elements that are in S1 or S2 (or in both).
Note: S1 [union] S2 is the same as S2 [union] S1.

The intersection of the sets S1 and S2 is the set S1 [intersect] S2, which contains the elements that are in BOTH S1 and S2.
Note: S1 [intersect] S2 is the same as S2 [intersect] S1.

The difference between the sets S1 and S2 is the set S1 / S2, which contains the elements that are in S1 and not in S2.
. Note. S1 / S2 IS NOT the same as S2 / S1.
Note. S1 / S2 is the same as S1 [intersect] [not]S2.

The symmetric difference between the sets S1 and S2 is the set S1 [symm diff] S2, which contains the elements that are in S1 and not in S2, or the elements that are in S2 and not in S1.
Note. S1 [symm diff] S2 is the same as S2 [symm diff] S1.
Note. S1 [symm diff] S2 is the same as (S1 [intersect] [not] S2) [union] (S2 [intersect] [not] S1).

The cartesian product of the two sets S1 and S2 is the set of all ordered pairs (a, b), where a [in] S1 and b [in] S2.

Linear Search Algorithm

I have published code that shows examples of the Linear Search Algorithm.

The linear search algorithm iterates through each item in our data structure in search for a specific value. If the current item matches, we can return, else we must continue to the next item.

In the worse case, this requires that we search through all items because in a unsorted structure, we cannot say whether an untesetd value is the value we are searching for.

Other Blogs that have covered this topic:
Dream.In.Code
Coding Bot

Examples of the Binary Search Algorithm

I have published code that shows examples of the Binary Search Algorithm.

In order for this algorithm to be applicable, we need to assume that we’re dealing with a sorted list to start. As a result, instead of proceeding iteratively through each item in the list, the binary search algorithm continually divides the list into two halves and searches each half for the element.

It can be shown that the maximum number of iterations this algorithm requires is equivalent to the number of times that we need to divide the list into halves. This is equivalent to a maximum number of iterations along the order of log2(n), where n is the number of items in the list.

Queue Data Structure

I have just published a program that shows examples of a queue data structure.

This page shows examples of the Queue Data Structure.

Queues operate under a property of First in First Out (FIFO), which is similar to waiting in a line.

The two main operations in a queue are to Enqueue (or insert an element) and Dequeue (or remove an element). Just like waiting in line, when an element is enqueued it is inserted at the back of the queue. And also like waiting in line, when an element is removed from a queue, it is removed from the front of the line.

Stack Data Structure

I have just published a program that shows examples of a stack data structure.

Stacks are an elementary Data Structure where all interaction with the data is done through the looking at the first element.

There are two main operations on Stacks, Push and Pop.

The push operation is used to insert an item into the stack. The name push comes from the fact that when an item is inserted into the stack it is put into the first element in the stack, so we think of it as lying on top of the stack.

Likewise, the pop operation is used to remove an item from the stack. Items are removed from the top of the stack first.