Set theory properties of relations

Question

Given the set $X=\{\{1\},\{2\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$

Show that the relation $$ \subseteq $$ on $X$ is reflexive, anti-symmetrical and transitive.

What do these properties mean in this context given that it's a set of sets?

I understand how it would be done if it were a set such as $X=\{(1,2),(2,1),(2,2)\}$ and so on.

Answer 1

It is a strange exercise but, anyway here are my two cents:

• Notice that the elements of $X$ are sets (subsets rather) contained in the set $\{1,2,3\}$

• $A$ and $B$ are related by "$\subseteq$" if and only if $x\in A$ implies that $x\in B$.

That the relation $ \subseteq $ has the desired properties follows by checking that

1. $A\subseteq A$ for set $A$ that appears in $X$.
2. $A\subseteq B$ and $B\subseteq A$ implies that $A=B$ for any $A$ and $B$ that appear in $X$
3. $A\subseteq B$ and $B\subseteq C$ implies that $A\subseteq C$ for all $A,B,C$ in $X$.

Properties 1-3 of course are valid not only for the elements of $X$ for any collection of subsets of a set.