Click here to view the embedded video.

A detailed tutorial on how to determine Dedekind cuts. Step by step tutorial including several examples of Dedekind cuts for reference.

A Dedekind cut is a partition of rational numbers into two non-empty sets A and B, such that all elements of A are less than elements of B, and A has no greatest element. The cut itself is a gap that is located between A and B, which is normally found by creating a new, irrational number, and setting it in the gap. What irrational number you use depends on what numbers you have partitioned into the two sets. It is like the number line of advanced algebra, that has both rational and irrational numbers on it instead of just integers. The Dedekind cut was named after Richard Dedekind.

Click here to view the embedded video.

A detailed tutorial on the pigeon-hole principle. Step by step tutorial including several examples of the pigeon-hole principle for reference.

The pigeon-hole principle is an important principle in math that states that if n items are to be put into m pigeon-holes, and n > m, then at least one pigeon-hole must contain more than one item. It is thought of as an extension of the counting principle. The pigeon-hole principle was first referred to as the drawer principle, or the shelf principle. Because of this, it is commonly called Dirichlet’s box principle or Dirichlet’s drawer principle. It is most commonly used with finite sets of elements; however, this principle can also be used with infinite sets.

Click here to view the embedded video.

A detailed tutorial on the definition of a finite set. Step by step tutorial including several examples of finite sets for reference.

There are many different types of sets, but one of the most common ones to run into a finite sets. A finite set is a set that has a finite number of elements – meaning a set with a definite number of elements, such as five, or ten. The number of elements in the set must be a natural number, and it is called the cardinality of a set. An empty set is considered to be finite, with a cardiality of zero, even though zero is not considered to be a natural number.

Click here to view the embedded video.

A detailed tutorial on disjoint sets. Step by step tutorial including several examples of disjoint sets and how to identify disjoint sets for reference.

A disjoint set is a term applied in set theory when two or more sets have no elements in common. For example, the sets {1, 2, 3} and {7, 8, 9} are disjoint sets because none of the numbers in the sets are the same. The formal way to say this is that two sets are disjoint sets if their intersection creates an empty set, in other words, nothing at all. An intersection is when you only take the values that are found in both sets. If none of the values are the same, this would be an empty set. Disjoint sets can be classified into further categories of piecewise, pairwise, or mutually disjoint provided that in a collection, at least two sets are disjoint.

Click here to view the embedded video.

A detailed tutorial on the union and intersection of sets. Step by step tutorial including several examples of the union and intersection of sets for reference.

Set theory is a branch of mathematics that deals with sets of numbers and the way that they interact with each other. One part of set theory is union and intersection. Union is represented by the symbol U, and means to combine the numbers in a set. The union of A and B states that for all x in A and B, the union contains all x in A and all x in B. Intersection is represented by an upside-down letter U, and means to only use numbers that are found in both sets. The intersection of A and B states that for all x in A and B, the intersection contains all x found in both A and B. The definitions might seem similar, but they are different.

Union:

A = {1, 2, 3, 4}, B = {2, 3, 6, 7}. The union would be {1, 2, 3, 4, 6, 7}.

Intersection:

A = {1, 2, 3, 4}, B = {2, 3, 6, 7}. The intersection would be {2, 3}

Click here to view the embedded video.

A detailed tutorial on using Venn diagrams in set theory. Step by step tutorial including several examples of how to use a Venn diagram in set theory for reference.

A Venn diagram is something we’ve all heard about before – probably in an english class. Venn diagrams are used as a visual representation for sets of different items, called elements. But it stands to reason that if you can use a venn diagram for sets, you can use them for set theory – a branch of mathematics that uses different sets of numbers. For set theory, they are used to display intersection and unions, and you can fill them out just like you would with a normal Venn diagram.

Click here to view the embedded video.

A detailed tutorial of set theory. Step by step tutorial including several examples of set theory for reference. Knowledge of set theory is required for most upper level math classes.

Set theory is the practice of sets and subsets. A set is a group of elements – numbers, items, anything. A set is expressed as A = {1, 2, 3, 4}, with A being the set, and anything inside the brackets being part of the set, being elements. A subset is also a set, but one that is the same as or contains part of another set. Each set has at least two subsets, because a subset can also be the exact same set, and an empty set. An empty set is expressed as a O with a line drawn through it, and it is a set that has no elements in it.

Click here to view the embedded video.

A detailed tutorial on the solving of combinations. Step by step tutorial including several examples of how to solve combinations for reference.

Combinations are often used with permutations. A combination is actually just the written representation of the permutation – with the permutation, you figure out how many different combinations there are, but with combinations you actually write down what those combinations are, not just how many there is. Many people prefer permutations because permutations are a lot less work. However, combinations do come up frequently, most notably in logic courses like discrete math.

Click here to view the embedded video.

A detailed tutorial on the solving of permutations. Step by step tutorial including several examples of how to solve permutations for reference.

A permutation is an ordered sequence of elements, also known as a set. Basically, a permutation is when you have a set amount of possibilities to be one thing (typically a variable), and then you have one less than that number of possibilities for your next variable, etc. Often you can use this to figure out exactly how many possible combinations in a set there are. Permutations are used very often in math, all done slightly different depending on the branch of mathematics, but it is first introduced in precalculus.