# How is the relation “the smallest element is the same” reflexive?

Let $$\mathcal{X}$$ be the set of all nonempty subsets of the set $$\{1,2,3,...,10\}$$. Define the relation $$\mathcal{R}$$ on $$\mathcal{X}$$ by: $$\forall A, B \in \mathcal{X}, A \mathcal{R} B$$ iff the smallest element of $$A$$ is equal to the smallest element of $$B$$. For example, $$\{1,2,3\} \mathcal{R} \{1,3,5,8\}$$ because the smallest element of $$\{1,2,3\}$$ is $$1$$ which is also the smallest element of $$\{1,3,5,8\}$$.

Prove that $$\mathcal{R}$$ is an equivalence relation on $$\mathcal{X}$$.

From my understanding, the definition of reflexive is:

$$\mathcal{R} \text{ is reflexive iff } \forall x \in \mathcal{X}, x \mathcal{R} x$$

However, for this problem, you can have the relation with these two sets:

$$\{1\}$$ and $$\{1,2\}$$

Then wouldn't this not be reflexive since $$2$$ is not in the first set, but is in the second set?

I'm having trouble seeing how this is reflexive. Getting confused by the definition here.

• Reflexive means that every element is related to itself. Thus, for reflexivity you have to consider one set only. Ok, we have that $\{ 1 \} \mathcal R \{ 1,2 \}$ but we have also $\{ 1 \} \mathcal R \{ 1 \}$ and $\{ 1,2 \} \mathcal R \{ 1,2 \}$ – Mauro ALLEGRANZA Apr 7 at 17:34
• Note: “reflexive” does not mean that if $x$ is related to $y$, then $x=y$. It means that if $x=y$, then $x$ is related to $y$. – Arturo Magidin Apr 7 at 17:44
• So it must be reflexive because both $A$ and $B$ belong to the same set $\mathcal{X}$? – qbuffer Apr 7 at 18:00
• @qbuffer Have a look at the updated version of my answer. – Haris Gusic Apr 7 at 18:49

Why are you testing reflexivity by looking at two different elements of $$\mathcal{X}$$? The definition of reflexivity says that a relation is reflexive iff each element of $$\mathcal X$$ is in relation with itself.
To check whether $$\mathcal R$$ is reflexive, just take one element of $$\mathcal X$$, let's call it $$x$$. Then check whether $$x$$ is in relation with $$x$$. Because $$x=x$$, the smallest element of $$x$$ is equal to the smallest element of $$x$$. Thus, by definition of $$\mathcal R$$, $$x$$ is in relation with $$x$$. Now, prove that this is true for all $$x \in \mathcal X$$. Of course, this is true because $$\min(x) = \min(x)$$ is always true, which is intuitive. In other words, $$x \mathcal{R} x$$ for all $$x \in \mathcal X$$, which is exactly what you needed to prove that $$\mathcal R$$ is reflexive.
You must understand that the definition of reflexivity says nothing about whether different elements (say $$x,y$$, $$x\neq y$$) can be in the relation $$\mathcal R$$. The fact that $$\{1\}\mathcal R \{1,2\}$$ does not contradict the fact that $$\{1,2\}\mathcal R \{1,2\}$$ as well.
A binary relation $$R$$ over a set $$\mathcal{X}$$ is reflexive if every element of $$\mathcal{X}$$ is related to itself. The more formal definition has already been given by you, i.e. $$\mathcal{R} \text{ is reflexive iff } \forall x \in \mathcal{X}, x \mathcal{R} x$$