Solving equations is computing pre-images

$$f^{-1}(y) = \left\{(x) \in X: f(x) = y \right\}$$

$$ f: X \longrightarrow Y $$

Rules

Fuck math. Let me say that a set $X$ is just a type X in C.

A composition law on $X$ is just a function $f$:

$$ \begin{align*} f: X \times X &\longrightarrow X\\ (x_0, x_1) &\longmapsto f(x_0, x_1) \end{align*} $$

The above is mathematical notation for what in C would be:

X foo(X x0, X x1);

Say $(X, f)$ and $(Y, g)$ are two set+compostion law pair.

Let $\phi$ a function $$\phi:X\longrightarrow Y$$ that would be

Y phi(X x);

We say that $\phi$ is a morphism if $$\phi(f(x0, x1)) = g(\phi(x0), \phi(x1))$$

X foo (X x0, X x1);
Y bar (Y y0, Y y1);
Y phi (X x);

/*
 * Then, if phi is a morphism: 
 * phi(foo(x0, x1)) == bar(phi(x0), phi(x1))
 *
 */

Total functions

A function is total if the output is well defined for all possible inputs. For composition laws, that is:

$$\forall x_0, x_1 \in X, f(x0, x1) \in X$$

In C we may want to use the errno for that:

errno = 0;
X x = foo(x0, x1);
// If foo is total: errno == 0 is always true;

Associativity

A composition law $f$ is associative if

$$f(f(x0, x1), x2) = f(x0, f(x1, x2))$$

foo(x0, foo(x1, x2)) == foo(x0, foo(x1, x2))

Identity

A right-identity is an element $e_r \in (X,f)$ such that

$$ f(x,e_r) = x; $$

A left-identity is an element $e_\ell \in (X,f)$ such that

$$ f(e_\ell, x) = x; $$

Division

A left-division, $a \setminus b$ is an elemnt $x \in X$ such that

$$ f(b, x) = a; $$

A right-division, $a / b$ is an element $x \in X$ such that

$$ f(x, b) = a; $$

Groups

A composition law $\star$ on a set $X$ is just a function

``

A group is a pair $(G, \star)$, where:

• $G$ is a set
• $\star$ is a composition law on $G$

That means

is such that

Examples

• $(\mathbb{Z}, +)$

$$(X,f,g)$$

$$g(x_0, f(x_1, x_2)) = f(g(x_0, x_1), g(x_0, x_2))$$

$$g(f(x_0, x_1), x_2) = f(g(x_0, x_2), g(x_1, x_2))$$