Confused about set theory terminology

Question

Hi I am having some confusion understanding some things.

In my class we were talking about relations and sets, and the professor gave this example. I am pretty much lost on understanding what is going on.

"Inclusion on P(x)"

$$P(\{1\})=\{\emptyset,\{1\}\} \subseteq \{(\emptyset,\emptyset),(\emptyset,\{1\}),(\{1\},\{1\})\}$$

Is reflexive, transitive, not symmetric, and is antisymmetric.

Basically I have no idea what is going on. I understood what a relation was and also what the four different properties are. I just don't understand what is going on here. What is the relation? Why are the now ordered pairs on a set on the right hand side?

Thanks for any clarification , I am hoping to understand this

Answer 1

Firstly, it isn't true that $\{\emptyset,\{1\}\} \subset \{(\emptyset,\emptyset),(\emptyset,\{1\}),(\{1\},\{1\})\}$, so this might have caused some confusion. However, both of these sets are important here, as we'll see.

The set we want to define a relation on here is $\mathcal{P}(\{1\})$, the power set of $\{1\}$ (which is $\{\emptyset,\{1\}\}$).

In this case, the relation is supposed to be $\subseteq$, so we want $(x,y)\in R$ exactly when $x\subset y$, where $x,y\in \mathcal{P}(\{1\})$. There aren't very many elements in $\mathcal{P}(\{1\})$, so we can state all of the inclusions: $${} \quad \quad \emptyset \subseteq \emptyset, \quad \emptyset \subseteq \{1\}, \quad \{1\}\subseteq \{1\}$$ Representing these as ordered pairs, we have $$R=\{(\emptyset,\emptyset),\quad (\emptyset,\{1\}),\quad (\{1\},\{1\})\},$$ which is exactly the set they listed above.

Answer 2

The equation does not make much sense as you've quoted it. It looks like what it is trying to say is something like:

Here is a concrete example on how set inclusion on $\mathcal P(X)$ is transitive, etc.:

Let's see what happens if we set $X=\{1\}$. Then $$ \mathcal P(\{1\}) = \{ \varnothing, \{1\} \} $$ and "inclusion" is the relation we usually notate with a $\subseteq$ symbol, which in this case is the set of pairs $$\bigl\{(\varnothing,\varnothing),(\varnothing,\{1\}),(\{1\},\{1\})\bigr\}$$ This relation is reflexive, transitive, not symmetric, and antisymmetric.