Sets
An introduction to sets. Some common sets are given as examples.
Working with Sets
The following concepts are introduced: subsets, the union of sets, the intersection of sets, the Cartesian product of sets, the cardinality of sets.
Definition of a Function
Two definitions of a function are given. The concepts of domain, codomain, range, and image are also explored.
Injective Functions
The definition of injective functions, a two-step approach to proving a function is injective, and plenty of examples.
Surjective Functions
The definition of surjective (onto) functions is given along with an outline of how to prove that a function is surjective.
Composition of Functions
The definition of function composition is given along with some examples.
The Identity Function
The identity function is introduced and some of its properties are examined.
Determining if a Function is Invertible
A function is invertible if and only if it is bijective.
Cardinality
The definition of cardinality is given and the method for proving that two sets have the same cardinality is provided, along with examples.
Definition of a Partition
Examples of partitions, followed by the definition of a partition, followed by more examples.
Definition of a Relation
The definition of a relation is given, along with a few examples.
Definition of an Equivalence Relation
The definition of an equivalence relation is given along with three examples.
Equivalence Classes
The definition of equivalence classes is given and several properties of equivalence classes are introduced.
Partitions and Equivalence Relations
A partition of a set determines an equivalence relation on that set. Also, an equivalence relation on a set determines a partition of the set.
Principle of Mathematical Induction
A proof of the Principle of Mathematical Induction using the Well-Ordering Principle.
The Division Algorithm
A proof of the division algorithm using the well-ordering principle.
Divisors
The definition of divisors is given along with a proof using the definition.
Greatest Common Divisor
The greatest common divisor is defined and the Euclidean Algorithm is used to calculate the gcd.
Linear Combinations
A procedure for writing the gcd of two numbers as a linear combination of the numbers is presented, along with an informal proof.
Relatively Prime Integers
The definition of relatively prime integers is introduced, along with two proofs.
Euclid’s Lemma
Both Euclid’s lemma and a generalization of the lemma are proved.
Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic is introduced along with a proof using the Well-Ordering Principle and a generalization of Euclid’s Lemma.
Least Common Multiple
The definition of the least common multiple is provided and is used to prove a couple of interesting results.
Congruence Modulo n
This video introduces the notion of congruence modulo n with several examples. In addition, congruence modulo n is shown to be an equivalence relation on the set of integers and the equivalence classes are described.
Integers Modulo n
The notion of congruence modulo n is used to introduce the integers modulo n. Addition and multiplication are defined for the integers modulo n.