I have to determine whether the $\subseteq$ relation is reflexive, symmetric, transitive and serial.

I'm slightly confused as to how to begin

I know that $A\subseteq B$ iff $\forall x (x\in A \rightarrow x\in B).$

And I have quantified definitions of reflexivity, symmetry, seriality, transitivity:

R is reflexive iff $\forall x Rxx$.
R is symmetrical iff $\forall x,y (Rxy \rightarrow Ryx)$.
R is transitive iff $\forall x,y,z (Rxy \land Ryz \rightarrow Rxz)$.
R is serial iff $\forall x \exists y Rxy.$

I understand these in the abstract. But I'm confused about how knowing the definitions of the relevant kinds of relations and the definition of $ \subseteq$ can help me determine whether $\subseteq$ is reflexive, symmetrical, transitive and/or serial.

Does anyone have any advice about how to proceed? (I've read through a similar question posted from 3 years ago but it does not help me understand how to go about doing this problem).

  • $\begingroup$ Apply the def... $A \subseteq A$ iff $\forall x (x \in A \to x \in A)$. $\endgroup$ – Mauro ALLEGRANZA Oct 4 '17 at 14:07
  • $\begingroup$ This sound good... thus: $\forall x (A \subseteq A)$ and the $\subseteq$ rel is reflexive. $\endgroup$ – Mauro ALLEGRANZA Oct 4 '17 at 14:08
  • $\begingroup$ And so on...... $\endgroup$ – Mauro ALLEGRANZA Oct 4 '17 at 14:09
  • $\begingroup$ You have to consider that the expression $x \subseteq y$ has the same "form" of $xRy$. It is a binary relation between sets. $\endgroup$ – Mauro ALLEGRANZA Oct 4 '17 at 14:27
  • $\begingroup$ Symmtery: in general no. If $A \subseteq B$, not necessarily we have also $B \subseteq A$ (otherwise $A=B$). $\endgroup$ – Mauro ALLEGRANZA Oct 5 '17 at 9:31

Reflexive: Is it true for all sets $A$ that $A\subseteq A$?

Symmetric: Is it always true that if $A\subseteq B$, then $B\subseteq A$?

Transitive: Is it always true that if $A\subseteq B$ and $B\subseteq C$, then $A\subseteq C$?


Instantiating your definitions, you have:

  • $\subseteq$ is reflexive if $\forall A,\ A \subseteq A$
  • $\subseteq$ is symmetric if $\forall A,B,\ A \subseteq B \Rightarrow B \subseteq A$
  • $\subseteq$ is transitive if $\forall A,B,C,\ (A \subseteq B \wedge B \subseteq C) \Rightarrow A \subseteq C$
  • $\subseteq$ is serial if $\forall A,\, \exists B,\, B \subseteq A$

You should use the given definition of $\subseteq$ to check which of these conditions hold and which do not.

I'll get you started with a proof that $\subseteq$ is reflexive. Let $A$ be a set. We need to prove $A \subseteq A$. For all $x$, the statement $x \in A \Rightarrow x \in A$ is certainly true, and hence $A \subseteq A$ is true. So $\subseteq$ is reflexive.


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