A Book of Abstract Algebra

Charles C. Pinter

Chapter 12 PARTITIONS AND EQUIVALENCE RELATIONS - all with Video Answers

Educators

Section 1

Problem 1

For each integer $r \in\{0,1,2,3,4\}$, let $A_{r}$ be the set of all the integers which leave a remainder of $r$ when divided by 5 . (That is, $x \in A_{r}$ iff $x=5 q+r$ for some integer $q$.) Prove: $\left\{A_{0}, A_{1}, A_{2}, A_{3}, A_{4}\right\}$ is a partition of $\mathbb{Z}$

Mohamed Mohamed

Numerade Educator

01:09

Problem 2

For each integer $n$, let $A_{n}=\{x \in \mathbb{Q}: n \leqslant x<n+1\}$. Prove $\left\{A_{n}: n \in \mathbb{Z}\right\}$ is a partition of $\mathbb{Q}$.

Narayan Hari

Numerade Educator

30:00

Problem 3

For each rational number $r$, let $A_{r}=\{(m, n) \in \mathbb{Z} \times \mathbb{Z}: m / n=r\} .$ Prove that $\left\{A_{r}: r \in \mathbb{Q}\right\}$ is a partition of $\mathbb{Z} \times \mathbb{Z}$

Oswaldo Jiménez

Numerade Educator

01:51

Problem 4

For $r \in\{0,1,2, \ldots, 9\}$, let $A_{r}$ be the set of all the integers whose units digit (in decimal notation) is equal to $r .$ Prove: $\left\{A_{0}, A_{1}, A_{2}, \ldots, A_{9}\right\}$ is a partition of $\mathbb{Z} .$

Mohamed Mohamed

Numerade Educator

01:37

Problem 5

For any rational number $x$, we can write $x=q+n / m$ where $q$ is an integer and $0 \leqslant n / m<1$. Call $n / m$ the fractional part of $x .$ For each rational $r \in\{x: 0 \leqslant x<1\}$ let $A_{r}=\{x \in Q ;$ the fractional part of $x$ is equal to $r\} .$ Prove: $\left\{A_{r}: 0 \leqslant r<1\right\}$ is a partition of $\mathbb{Q}$.

Prashant Bana

Numerade Educator

01:56

Problem 6

For each $r \in \mathbb{R}$, let $A_{r}=\{(x, y) \in \mathbb{R} \times \mathbb{R}: x-y=r\} .$ Prove: $\left\{A_{r}: r \in \mathbb{R}\right\}$ is a partition of $\mathbb{R} \times \mathbf{R}$

Prathan Jarupoonphol

Numerade Educator