In normal, everyday language, it is quite common to hear people speak with conditionals. The problem with everyday language is that it is not precise. Specifically, people tend to use implications when speaking, but sometimes what a person is implying is really a biconditional.

When writing mathematics, it is imperative to be as precise as possible. As such, if what you want to state involves a biconditional, you should use the double-ended arrow ↔. It is almost never appropriate to use an implication arrow → when a biconditional is needed.

The exception to the above rule is definitions. The purpose of a definition is to assign a meaning, idea, or concept to a single word. Hence, it is usually OK to state definitions in terms of implications.

Suppose we want to define some new word, which we’ll refer to simply as word, and we want to assign a definition to word, which we’ll refer to simply as definition. The most accurate way to convey meaning is by saying

word ↔ definition

Remember that for any two propositions p and q, we have that

p ↔ q ⟺ (p → q) ∧ (q → p).

Hence, instead of saying word ↔ definition, we would be fine to instead say

word → definition

or

definition → word.

because even though an implication is used, what is really meant is the biconditional.

Here, we present some mathematical words and definitions which you’ll likely be familiar with already, but they are being presented here just for the sake of completeness.

Notice in the above definition, when writing out the logical quantified statements, we didn’t use a single letters to represent the propositions. Instead, we just wrote out the propositions in English.

Also note that we wrote the “and only if” part in parenthesis. This is because here, we want to stress that in a definition, biconditionals are used. Again, it is usually fine to to just say the “if” part in definitions because what’s really meant is the biconditional. For the rest of this section, we will continue to write the “and only if” part in parenthesis. In future sections, it may be excluded, fully written out, or included in parenthesis.

It may seem a bit excessive to go over the details of how even and odd integers are defined in a college-level set of notes. The point of the examples is not to teach what even and odd integers are. The point of the above examples is to show how a mathematical definition can carry a lot of nuance, even for a definition as simple as even or odd. Additionally, by using commonly known terms, we get a chance to translate their definitions into a more mathematical form.

Notice that in our definition of even, there are three basic requirements that have to be satisfied in order for a number n to be classified as even:

n must be an integer
n = 2k for some number k
k must be an integer.

If any combination of these three requirements fail, then we cannot describe n as even. In mathematics, we require utmost precision in order to know exactly what idea is being conveyed.

Three requirements must also be met for the number n to be called odd:

n must be an integer
n = 2k + 1 for some number k
k must be an integer.

We can write this definition out using the mathematical notation we’ve developed so far, and write out an equivalent English sentence:

e(n):	n is even
o(n):	n is odd
a(n, k):	n = 2k
b(n, k)	n = 2k + 1

Remember that the universe of discourse for symbols n and k in the two quantified statements above is ℤ, the universe of all integer numbers.

All definitions in mathematics work this way, they assert conditions that must be satisfied before the associated word can be applied.

Again, let’s take a careful look at the definition of parity. In order for the word parity to be applied, we first need two integers, let’s just use m and n for the time being. If m is not an integer, or n is not an integer, then we can’t use the word parity in reference to m and n.

If we wanted to use a more notational style in defining parity, we could do the following:

p(a, b):	a and b have the same parity
e(a):	a is even
o(a):	a is odd

∀m, n [p(m, n) ↔ (e(a) ∧ e(b)) ∨ (o(a) ∧ o(b)) ]

However, suppose a different number p was an integer, then we could use the word parity in reference to numbers m and p since both m and p are integers.

Here is one more definition which will be used quite frequently.

In the previous two sections, we examined arguments that had universally quantified statements as premises, however, we were concerned more with what form the conclusion of those arguments took. We never discussed how we could get universally quantified statements as premises. Appealing to definitions is one way to do so. We’ll see examples of this in the next section.

Whereas definitions are things that can be decreed, axioms represent something fundamentally different. In some sense, axioms are statements of mathematical interest that are intuitively correct, but require no proof of correctness.

Something worth noting is that when we assert some collection of axioms, we are essentially creating a branch of mathematics. Everything we do with those axioms develop that branch of mathematics into something more mature. If even one of those axioms is altered, then everything deduced from the original axioms no longer holds. Instead, you have an all new branch of mathematics, with new results, and even more exciting discoveries to be made!

Whereas axioms are statements of mathematical interest that are asserted, and define a entirely new branch of mathematics, theorems are always deducible from axioms and definitions.

Axioms can only ever appear as premises in arguments, theorems can be premises or conclusions of arguments.

Typically what happens is that we start with a collection of axioms and definitions. From those initial axioms and definitions, we can deduce some initial collection of theorems. Next, we deduce a second round of theorems using the previous theorems, axioms and definitions. We can in theory repeat this process to yield ever more theorems.

During the course of all of this theorem proving, we may even come up with new definitions for a variety of reasons.

Notice from the above discussion we noted that some theorems can be proven from other, previously deduced theorems in addition to the given axioms and definitions. In other cases, some theorems are simply special cases of other theorems. We have special names names for those too.

Oftentimes, the word theorem is reserved for major results. We could also be pedantic in describing various kinds of theorems as lemmas and corollarys.

Most of the time, it’s not important to correctly classify certain theorems as lemmas or corollarys. The important thing about them is that they are all theorems that can be deduced from the axioms, definitions, and other deduced theorems.

The final mathematical building block we’ll discuss is the proof.

Up to this point in this chapter, we’ve discussed arguments at length, especially how to determine if a given argument is valid. We also touched lightly on the subject of using the rules of inference to chain together logical implications to yield new logical implications. All a proof is is just a valid argument.

Typically, we describe a proof as being given in reference to a theorem. By this, what we mean is that if someone proposes a statement of mathematical interest, if a proof can be given for that statement, then that statement is henceforth called a theorem (because it can be deduced.)

Remember that an argument is simply an implication, with a conjunction of multiple propositions as the hypothesis, and a single proposition as the conclusion. Any of the propositions in the hypothesis or conclusion can be primitive or compound.

In the next section, we start learning specific methods of proof, and of ways to devise proofs. This is the primary activity of mathematics.