Equivalence relations. Factorization of mappings

Show that the quotient set $\mathbb {R}^{2} / \sim$ obtained from Fig. 8, and any line $l$ intersecting the axis $Ox$, are in bijective correspondence.

Fig. 8

Set $P(x, y) \sim P\left(x^{\prime }, y^{\prime }\right)$ for points of the real coordinate plane $\mathbb {R}^{2}$ precisely when $x^{\prime }-x \in \mathbb {Z}$ and $y^{\prime }-y \in \mathbb {Z}$. Prove that $\sim$ is an equivalence relation and that the quotient set $\mathbb {R}^{2} / \sim$ is geometrically represented by points on a torus (the surface of a doughnut; Fig. 11).

Fig. 11

Show that sets of two, three, and four elements have, respectively, 2, 5, and 15 distinct quotient sets.

Let $\sim$ be an equivalence relation on a set $X$ and $f: X \rightarrow Y$ a map for which

Show that this compatibility condition of $f$ with $\sim$ (weaker than the one considered in item 2) allows one to correctly define the induced map $\bar{f}: \bar{x} \mapsto f(x)$ from $X / \sim$ to $Y$, leading to the factorization $f=\bar{f} \cdot p$, but $\bar{f}$ need no longer be injective. What is the condition for injectivity of $\bar{f}$?

Depict by diagrams the partially ordered sets:

1. $\mathcal{P}(\left\{ a, b, c, d\right\} )$;

2. the set of all divisors of the integer 24 (the order relation is indicated in the text).

The order relations indicated in the text (item 4 of § 6): on the set $\mathcal{P}(S)$ of subsets of a set $S$, the usual inclusion relation $R \subset T$ between subsets; on the set of natural numbers, the relation $d \mid n$ ($n$ is divisible by $d$). One says that $y$ covers $x$ if $x<y$ and there is no $z$ with $x<z<y$; in a diagram the elements are drawn as points, and if $y$ covers $x$, then $y$ is placed above $x$ and $x$ is joined to $y$ by a straight segment.