MHB What is the Equivalence Class for a Fixed Integer in Hurricane Lane's Aftermath?

[karush]

View attachment 8332
ok need help with these 3 questions

I know its fairly easy but its new to me, so

we have had hurricane Lane here this week
but mahalo much

 

[Olinguito]

13. Showing that the given relation is an equivalence relation is straightforward. I suppose you are stuck at describing the equivalence classes. Let $a$ be a fixed integer. What is the equivalence class containing $a$, i.e. $[a]$? Well, $b\in[a]$ $\iff$ $b\sim a$ $\iff$ $b-a=n\in\mathbb Z$ $\iff$ $b=a+n$ $\iff$ $b\in\{a+n:n\in\mathbb Z\}$. Hence $[a]=\{a+n:n\in\mathbb Z\}$.

14. Is this relation transitive? Hint: $1\cdot0\ge0$ and $0\cdot(-1)\ge0$.

15: Suppose $\frak P$ is a partition of a set $S$; then the relation is $a\sim b$ iff $a$ and $b$ belong to the same subset of $S$ in $\frak P$. By the definition of a partition, every member of $S$ belongs to some subset of $S$ in $\frak P$; hence ~ is reflexive. It is clearly symmetric. I’ll leave you to show that it is transitive, hence an equivalence relation.

 

[karush]

13. Showing that the given relation is an equivalence relation is straightforward. I suppose you are stuck at describing the equivalence classes. Let $a$ be a fixed integer. What is the equivalence class containing $a$, i.e. $[a]$? Well, $b\in[a]$ $\iff$ $b\sim a$ $\iff$ $b-a=n\in\mathbb Z$ $\iff$ $b=a+n$ $\iff$ $b\in\{a+n:n\in\mathbb Z\}$. Hence $[a]=\{a+n:n\in\mathbb Z\}$.

14. Is this relation transitive? Hint: $1\cdot0\ge0$ and $0\cdot(-1)\ge0$.

15: Suppose $\frak P$ is a partition of a set $S$; then the relation is $a\sim b$ iff $a$ and $b$ belong to the same subset of $S$ in $\frak P$. By the definition of a partition, every member of $S$ belongs to some subset of $S$ in $\frak P$; hence ~ is reflexive. It is clearly symmetric. I’ll leave you to show that it is transitive, hence an equivalence relation.

that was very helpful ;)