Today we discussed equivalence statements and relations further.
Recall that relation statements can be written as follows:
We then discussed equivalence classes which are ways of writing relations. This can be notated as follows:
![[\mathrm{x}] : = \{\mathrm{a}\in\mathrm{A}: \mathrm{aRx}\subseteq\mathrm{A}\}](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bx%7D%5D+%3A+%3D+%5C%7B%5Cmathrm%7Ba%7D%5Cin%5Cmathrm%7BA%7D%3A+%5Cmathrm%7BaRx%7D%5Csubseteq%5Cmathrm%7BA%7D%5C%7D&bg=ffffff&fg=444444&s=0&c=20201002)
When something is an equivalence relation, the equivalence classes can be notated as:
![\mathrm{A} = \bigcup_{\mathrm{x}\in\mathrm{A}}=[\mathrm{x}]](https://s0.wp.com/latex.php?latex=%5Cmathrm%7BA%7D+%3D+%5Cbigcup_%7B%5Cmathrm%7Bx%7D%5Cin%5Cmathrm%7BA%7D%7D%3D%5B%5Cmathrm%7Bx%7D%5D&bg=ffffff&fg=444444&s=0&c=20201002)
this equivalence classes are disjoint when:
![[\mathrm{x}]\cap[\mathrm{y}]\neq{0}](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7Bx%7D%5D%5Ccap%5B%5Cmathrm%7By%7D%5D%5Cneq%7B0%7D&bg=ffffff&fg=444444&s=0&c=20201002)
![\Leftarrow\Rightarrow[\mathrm{x}] = [\mathrm{y}]](https://s0.wp.com/latex.php?latex=%5CLeftarrow%5CRightarrow%5B%5Cmathrm%7Bx%7D%5D+%3D+%5B%5Cmathrm%7By%7D%5D&bg=ffffff&fg=444444&s=0&c=20201002)
We then got our fun math fact of the day. Since Pac-Man can go from the right-side to the left-side and from the top to the bottom, you can assume that the grid layout looks like this:
This means that if you fold the grid so the corresponding sides match up, it becomes clear that Pac-Man lives in a doughnut 🙂
Peace
Emily
Triplet Integral 