###### (Originally written for my Intro to Homological Algebra Summer Seminar Group at SJC)

One of the first things you learn in Algebra, is how to manipulate equations.  Given the equation:

$2x + 3 = 4,$

we quickly learned how to subtract 3 from both sides of the equation.  In this post, I would like to go through the very formal definitions which make this possible.

### The Definition of a Group and Why We Can Cancel

Recall that a group is a set, $B$, together with a binary operation $B \times B \xrightarrow{\mu} B$, which is associative, has a unique identity, and has unique inverses.  Before going forward, it is important to understand what it means to have a function, $\mu$, and what it means to have pairs in $B \times B$ as its input.  For a quick review of some foundations for this post, check out the section Review: Sets, Cartesian Products, Equality, and Functionsbelow.

OK, once you feel like you understand Cartesian Products and functions, let’s say what addition is!  So addition on a set, $B$, by the definition of a group, is a binary operation, i.e. a function,

$B \times B \xrightarrow{\mu} B$

Let’s unpack this.  So for two elements, $p$ and $q$ in the set $B$, we have a way to add them together to produce a third element, call it $y \in B$:

$\mu(p,q)=y$

but we normally just write:

$p+q=y$

OK, but let’s look at what it means to be a function, and what this has to do with the whole point of this post!  Recall we started with the equation

$2x + 3 = 4$

Well suppose that the set, $B$ is the set of all of our numbers.  Then $2x+3$ and $4$ are two elements in $B$, but that are equal.  Two other numbers that are equal are $-3$ and $-3$.  Therefore, we have two ways to write the same pair:

$(2x+3, -3)$ and $(4, -3)$

Since their first components are equal and their second components are equal then we have

$a = (2x+3, -3) = (4, -3)$

Now, remember our function, $\mu: B \times B \to B$?  For the moment let’s call $B \times B$ the set $A$, so that $A = B \times B$.  Well then being a function, $\mu : A \to B$,  means that  if $(a, b_1) , (a,b_2) \in \mu$ then we must have that $b_1 = b_2$.  Note that for our setup, we have

$b_1 = \mu(a) = (2x+3)+(-3)$ and $b_2 = \mu(a) = (4) + (-3)$

But since $b_1 = b_2$, then we must have that

$(2x+3) + (-3) = (4) + (-3)$

So now you finally know that the reason you can add to both sides is that addition is a function!!!  Anyway, now you could use the associative property of our group,

$(2x)+(3 + (-3)) = 4 + -3$

and then the additive inverse property,

$(2x)+0 = 4 + -3$

and now the identity property,

$(2x) = 4 + -3$

to finally see why we can cancel something from both sides of an equation.

Concept Check: You should feel free to move on from the ideas in this post when you can easily understand and show that, similarly, we can cancel the $4$ being multiplied on the right hand side of the equation

$2x + 3 = x \cdot 4$

### Review: Sets, Cartesian Products, Equality, and Functions

Assuming we have basic understanding of set theory, we need to first consider the set of all pairs of elements from a set $A$ and a set $B$, which is called the Cartesian Product, and written $A \times B$.  More formally, we define the Cartesian Product by:

$A \times B := \{ (a, b) \ \vert \ a \in A \text{ and } b \in B \}$

Note that the only way two pairs $(a,b)$ and $(x,y)$ are allowed to be equal is if both $a=x$ and $b=y$.  As it turns out, in order to say what a function is, we will need Cartesian products.

First we define a relation, which is simply a subset of a Cartesian Product.  That is to say, a relation, $R$, between $A$ and $B$, is simply some subset

$R \subset A \times B$

For example, if I consider the set of all teams participating in the world cup this year, call it $T$, and the set of all locations of the stadiums being used in the world cup this year, call it $L$, then I can consider the relation of locations in which certain teams have played during the tournament this year as the subset $A \subset T \times L$.  So for some examples:

• (Egypt, Ekaterinburg), (Egypt, St. Petersburg), (Senegal, Moscow), (Portugal, Sochi), and (Russia, Moscow) are all elements of $A$ while
• (Iran, Moscow) is not an element in $A$, since Iran did not play in Moscow during this World Cup (I looked it up).

Notice how for my relation, $A$, it allows for both the $T$-values and $L$-values to repeat (you would often call these the $x$-values and $y$-values, respectively, in your experience).  Since you were told presumably that for a function, the first component is not allowed to repeat, then you have just observed that not all relations are functions.  A function is a special type of relation, that is:

function $f: A \to B$ is a relation, $f \subset A \times B$, in which if both $(a,b)$ and $(a, b')$ are in the set, $f$, then it must be that $b = b'$.

Note that usually you simply write an element $(a,b) \in f$ as: $f(a) =b$.  But it’s important to realize that technically, the definition using the pairs is the more careful and formal one.

Concept Check: You should feel free to move on from this definition of a function using Cartesian Products if you can explain why this definition implies the vertical line test which we were all taught when we first learned to distinguish functions from non-functions.

OK, now you are ready to move back to the original discussion.