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Im working on the book "Introduction to topology and modern analysis - by G.F.Simmons" where I have stumbled upon an exercise Im not fully comprehend, since no proper defintion of relations was given so far. Im wondering whether the question is self explanatory.

Let $U$ b the set $\{1,2\}$. There four subsets. List them. If $A$ and $B$ are arbitrary subsets of $U$, there are $16$ possible relations of the form $A \subseteq B$. Count the number of true ones.

What I not really understand is "true" in that regard.

I do understand that the power set $P(\{1,2\}$) consists of $\{\emptyset,\{1\},\{2\},\{1,2\}\}$. Also I know that there are $2^4$ possible relations since here are $2 \cdot 2 = 4$ pairs of one element from $A$ and one from $B$. However, I do not understand what is considered to be a true relation(total, equivalence relation...) Appreciate any insights!

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The word "relation" here is not being used in its usual technical sense (a relation between $X$ and $Y$ is a subset of $X\times Y$). Rather, it's just saying that there are 16 expressions of the form $A\subseteq B$ that you can write down where $A$ and $B$ are subsets of $U$. This is because there are $4$ choices for $A$ ($4$ elements of $\mathcal{P}(U)$) and $4$ choices for $B$.

For example, here are three possibilities:

  1. $\emptyset\subseteq \{1,2\}$
  2. $\{1\} \subseteq \{2\}$
  3. $\{1\} \subseteq \emptyset$

Now the problem is just asking you to count how many of these 16 expressions are true. In my examples, 1 is true, while 2 and 3 are false.

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  • $\begingroup$ While I agree that in context of this question the "relation" $A \subseteq B$ is not a subset of $X \times X$, so it is not a relation between $X$ and itself, nonetheless it is a subset of $P(X) \times P(X)$, hence it is a relation between $P(X)$ and itself. $\endgroup$ – Lee Mosher Nov 1 '17 at 21:06
  • $\begingroup$ @LeeMosher Sure, of course $\subseteq$ is a binary relation on $P(X)$. But then it's $\subseteq$ which is the relation, not $A\subseteq B$ for particular sets $A$ and $B$. There certainly are not 16 relations here, and it doesn't make any sense to ask how many of them are true. So I think G.F. Simmons was simply not thinking about this meaning of the word "relation" when he wrote the problem. $\endgroup$ – Alex Kruckman Nov 1 '17 at 21:30

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