The relation R = {(x, y)|x − y is an integer} is an equivalence relation on the set of rational numbers.
I'm kind of confused with this question and what it is asking me to do. In order to solve this, do I just take two arbitrary rational numbers and disprove that x-y is an integer? For example, 1/2 - 2/3 is not an integer. How does this tie in with an equivalence relation?
Thanks in advance.
On
There are three properties to show $R$ is an equivalence relation: (i) $x \sim x$ because $x-x=0\in\mathbb Z$. (ii) If $x-y\in\mathbb Z$ then $y-x\in\mathbb Z$. (iii) And finally if $x-y\in\mathbb Z$ and $y-z\in\mathbb Z$ then $x-z = (x-y)+(y-z)\in\mathbb Z$. Therefore $R$ is an equivalence relation.
The class of $0$ is $\mathbb Z$ and the class of $\frac12$ is $\{\frac12+n | n\in\mathbb Z\}$.
You are given the relation $R=\{(x,y)|x-y\in\mathbb{Z}\}$ on the rational numbers, and are asked to prove that it is an equivalence relation. What is an equivalence relation? It is a relation that has the following three properties, and you must prove that each of these properties are true.
1. Reflexive. This means that for any rational number $x$, it must be true that $(x,x)\in R$. For your particular relation, take some arbitrary rational number $x$, and look at the definition of $R$. If it is to be true that $(x,x)\in R$, then $x-x$ must be an integer. Is this so?
2. Symmetric. That your relation is symmetric means that if you have two rational numbers $x$ and $y$, and if $(x,y)\in R$, then it must be true that $(y,x)\in R$. For your problem, this means that if $x-y$ is an integer, then $y-x$ must also be an integer. Can you prove this?
3. Transitive. For $R$ to be transitive, it must hold that whenever $x$, $y$, and $z$ are rational numbers such that $(x,y)\in R$ and $(y,z)\in R$, then it must be true that $(x,z)\in R$. This means that if $x-y$ is an integer, and $y-z$ is an integer, then $x-z$ must also be an integer. Is this true for your relation? This part might be tricky, so as a hint, you can consider the sum $(x-y)+(y-z)$.