In this module you will learn the most important and fundamental proof technique called the Forward-Backward Method, which consists of both a Forward Process and a Backward Process. You will see the details of how both of these processes are used and you will be exposed to a first example of how to read someone else’s proof. It is critical that you be able to use the Forward-Backward Method because all the remaining proof techniques are explained in terms of this method.

In this module, the goal is to learn how to use mathematical definitions in both the forward and backward processes of the Forward-Backward Method. Learning to do this correctly is critical because virtually every proof in the rest of the course relies on your ability to do so. Some common mathematical terminology associated with proofs is also introduced at the end of this module.

In this module, you will learn that all proof techniques other than the Forward-Backward Method have “key words” associated with them, meaning that, if you see those key words in either the Forward or Backward Processes, you might consider using the associated proof technique. This provides an approach for starting a proof, namely, looking for key words in the hypothesis and conclusion of the statement you are trying to prove (if you do not find any key words, then you might start the proof with the Forward-Backward Method). You will also learn about the Construction Method, which is a proof technique you should consider using when you see the key words “there is” (“there are,” “there exists”) in the Backward Process. You will also be exposed to reading someone else’s proof that uses the Construction Method.

In this module, you will learn the Choose Method, which is a proof technique you should consider using when you see the key words “for all” (“for each,” “for every,” “for any,”) in the Backward Process. You will also be exposed to reading someone else’s proof that uses the Choose Method.

In this module, you will learn Specialization, which is a proof technique you should consider using when you see the key words “for all” (“for each,” “for every,” “for any,”) in the Forward Process. You will also be exposed to reading someone else’s proof that uses Specialization.

All the statements so far have contained at most one quantifier (either “there is” or “for all”). In this module, you will learn how to prove statements that involve more than one quantifier (referred to as nested quantifiers). You will also see an example of reading someone else’s proof that involves nested quantifiers

In this module, you will learn how to write the negation of a statement, that is, which it means to say that a statement is not true. Rules are also provided for writing the negation of a statement that contains quantifiers. Writing the negation of a statement is a skill that is needed in the next two proof techniques.

In this module, you will learn the Contradiction Method, which is a proof technique you should consider using when you see the key words “no” or “not” in the Backward Process. In fact, you can you this technique to prove that any statement is true. You will also see the advantages and disadvantages of using this technique when compared to the Forward-Backward Method. You will also be exposed to reading someone else’s proof that uses the Contradiction Method.

In this module, you will learn the Contrapositive Method, which is a special proof by contradiction. This method is compared to both the Forward-Backward Method and the Contradiction Method to highlight advantages and disadvantages of each technique. You will also be exposed to reading someone else’s proof that uses the Contrapositive Method.

In this module, you will learn the Uniqueness Methods, which are proof techniques you should consider using when you see the key words “unique” or “one and only one” in either the Forward Process or the Backward Process. In particular, you should consider using the Forward Uniqueness Method when you know that there is a unique object with a certain property such that something happens. In contrast, you should consider using the Backward Uniqueness Method when you have to show that there is a unique object with a certain property such that something happens.

In this module, you will learn Induction, which is a proof technique you should consider using when you want to show that some statement is true for all values of an integer n = 1, 2, … You will also see some variations of this method and learn to read a proof that uses Induction.

In this module, you will learn the Either / Or Methods, which are proof techniques you should consider using when you see the key words “either / or” in either the Forward Process or the Backward Process. In particular, you should consider using a Proof-By-Cases when you see these key words in the Forward Process and a Proof-By-Elimination when you see those key words in the Backward Process.

In this module, you will learn the Max / Min Methods, which are proof techniques you should consider using when you see the key words “max” or “min” in either the Forward Process or the Backward Process. The main idea is to convert statements containing these key words into equivalent quantified statements so that you can then use the Construction, Choose, and Specialization Methods.

A summary of all the proof techniques is given in this module. A final example of reading a proof that uses many of these techniques is used to bring together your knowledge of how to read and do proofs.