Sets, Proofs, and the Axiomatic Method

Axiomatic method

Linear Algebra is about vectors. You would think that the theory begins with a definition of what a vector is, but in mathematics we don't do that. It turns out that there are many different mathematical objects that you might want to call a "vector", and people wouldn't agree on which definition is the right definition of vector. So instead of trying to define what a vector is we lay down rules for what we should be able to do with "vectors," and then we figure out the consequences of those rules; we call those consequences "theorems." Then anyone who has a collection of mathematical objects in mind that they would like to call vectors only has to check that their "vectors" satisfy our rules, and then they know that all the consequences ("theorems") we found will also apply to their "vectors." This is the axiomatic method. The rules we decide on are the axioms. The "consequences" we find are the Theorems.

In calculus you probably saw a description of vectors as two or three dimensional arrows. This description comes from physics, and you can find a much more detailed and physically relevant version of that description in the Feynman lectures on Physics (vol.I, ch11). These calculus/physics vectors are an example of vectors in linear algebra, but there are many more examples.

Set theory

The most common way to discuss mathematical objects is to use the theory of sets.

Here are some a very quick summary of the vocabulary of set theory.

• A set is a collection of mathematical objects.
• If $A$ is a set and $x$ is some other mathematical object, then $x\in A$ means that $x$ is one of the objects that belong to $A$. If $x\in A$ then $x$ is an element of $A$.
• Two sets are equal if they contain the same elements: if $A$ and $B$ are sets then $A=B$ means that for every $x\in A$ one has $x\in B$, and for every $$x\in B$$ one has $x\in A$.
• There is one set that contains no elements. It is called the empty set.
  The symbol for the empty set is "$\varnothing$" (or "$\{\}$", or "$\emptyset$")

Notation to specify a set

If a set only has a few elements you can describe the set by listing its elements, as in

$$A=\{1, 3, 2, 4\}, \text{ or } B=\{a, b, P\}$$

If the set contains a sequence of elements you can write

$$A=\{x_1, x_2, \dots, x_n\}$$

and hope that the reader knows what you mean by "$\dots$". You can even do this with certain infinite sets, e.g. you can write

$$V = \left\{\frac12, \frac14, \frac18, \frac1{16}, \dots \right\}$$

and hope your reader recognizes the pattern.

A very common way of specifying a set uses this notation:

$$A = \{x \mid x\text{ satisfies }\mathcal{C}\} \text{ or } A = \{x : x\text{ satisfies }\mathcal{C}\}$$

This defines $A$ as the set of all mathematical objects $x$ that statisfy the condition $\mathcal{C}$. For example, the set $V = \{\frac12, \frac14, \frac18, \frac1{16}, \dots \}$ above could also be written as

$$V=\left\{\frac{1}{2^n} \,\Big|\, n\in \N\right\}.$$

Your reader has to know that $\N = \{1, 2, 3, 4, \dots\}$ is the set of natural numbers.

Another variant:

$$B = \{x\in A \mid x\text{ satisfies }\mathcal{C} \}$$

This says that $B$ is the set of all mathematical objects $x$ that belong to the set $A$ and that satisfy the condition $\mathfrak{C}$.