Determining if this set is an equivalence relation

Question

I have the following here:

For the following sets $A$ and binary relations $\sim$, decide of $\sim$ defines an equivalence relation on $A$ or not.

a)
Set: $A=\mathbb{R}$
Relation: $x \sim y$ if $x=ay$ for some $a \in \mathbb{Q}$

b)
Set: $A=\mathbb{R}$
Relation: $x \sim y$ if $x=ay$ for some $a \in \mathbb{Q}\backslash \{0\}$

I know I have to use the reflexive, symmetric and transitive properties here.

I know that:

Reflexive means that for all $a\in A$, $a \sim a$

Symmetry means that for all $a,b\in A$, if $a \sim b$, then $b \sim a$

Transivity means that for all $a,b,c\in A$, if $a \sim b$ and $b \sim c$ then $a \sim c$.

I understand the definitions and if the question just gave a bunch of elements in sets, I could do this without issues. How do I use the properties here though? Can someone help me out here?

Answer 1

I know that:

Reflexive means that for all $a\in A$, $a \sim a$
Symmetry means that for all $a,b\in A$, if $a \sim b$, then $b \sim a$
Transivity means that for all $a,b,c\in A$, if $a \sim b$ and $b \sim c$ then $a \sim c$.

Correct, but to avoid confusion with the definitions, let us use $x,y,z$ and also less words and more logic symbols.

Reflexive: $\forall x\in A: x \sim x$
Symmetry: $\forall x,y\in A: (x \sim y\to y \sim x)$
Transivity: $\forall x,y,z\in A: (x \sim y\land y \sim z\to x \sim z)$.

Next, replace $A$ and $\sim$ with the given definitions

a)
Set: $A=\mathbb{R}$
Relation: $x \sim y$ if $x=ay$ for some $a \in \mathbb{Q}$

Reflexive: $\forall x\in\Bbb R:(\exists a\in\Bbb Q~. x = ax)$ ?
Symmetry: $\forall x,y\in \Bbb R:((\exists a\in\Bbb Q~.x = ay)\to (\exists b\in\Bbb Q~.y = bx))$ ?
Transivity: $\forall x,y,z\in\Bbb R:((\exists a\in\Bbb Q~.x = ay)\land(\exists b\in\Bbb Q~. y = bz)\to(\exists c\in\Bbb Q~. x =cz))$ ?

Finally, do these claims hold up to scrutiny?

Well for every real number ($x$) there does exists a rational number ($a$) where $x=ax$; because $1$ is a rational number. Therefore this relation is reflexive over the real numbers. Symmetry and Transitivity are investigated similarly.

b)
Set: $A=\mathbb{R}$
Relation: $x \sim y$ if $x=ay$ for some $a \in \mathbb{Q}\smallsetminus\{0\}$

Clearly you just need to consider how the answers above would change when none of the rational numbers may be zero. Think carefully.