Smallest and biggest symmetric relations Can anyone give me some hints about this homework? I`m really stuck. Thank you.
Let R is a binary relation in A. Define T and S using R, such that S is the smallest symmetric relation which $R \subseteq S$ and T is the biggest symmetric relation  which $T \subseteq R$.
(In other words, S and T are binary relations, defined using the sets R,$\varnothing, Id_{A}, A \times$ A and the operations $\cup,\cap , \ ^{-1}, \circ $)
 A: For $T$, first ask yourself what ordered pairs in $R$ absolutely need to be removed to make a symmetric relation; for $S$, what ordered pairs in absolutely have to be added to $R$ to make a symmetric relation. The problem then is to use the allowable tools to remove them. 


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*What kind of relation $P$ on $A$ has the property that $P=P^{-1}$?

*If $P$ is a relation on $A$, what does $P\cap P^{-1}$ look like? Answering this should help you with $T$.

*Ask and answer a question similar to (2) that helps with $S$.
There is an alternative way to get at $S$. It’s often the case that the smallest whatsit containing a given object is the intersection of all of the whatsits containing it; this works whenever the intersection of whatsits is again a whatsit. Can you see why it works? (Note that this is a pretty common construction, so it’s well worth understanding.)
In this case you want $S$ to be the smallest symmetric relation on $A$ containing $R$, so in your problem a whatsit is a symmetric relation on $A$. Answering the following questions should help you with $S$.


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*Is it true that the intersection of symmetric relations on $A$ is symmetric?

*Is there at least one symmetric relation on $A$ that contains $R$?
A: General Hint. Can you describe the property "$U$ is a symmetric relation" using $\cap$, $\cup$, ${}^{-1}$, and/or $\circ$?  Note that "$U$ is symmetric" if and only if $(a,b)\in U\Longleftrightarrow (b,a)\in U$ and that $(x,y)\in U\Longleftrightarrow (y,x)\in U^{-1}$. 
Hints for $S$. If $S_1$ and $S_2$ are symmetric relations, and both contain $R$, is $S_1\cap S_2$ a symmetric relation that contains $R$? Is there at least one symmetric relation that contains $R$ (not necessarily the smallest one...)
Hints for $T$. What should you remove from $R$ to get something symmetric? What should you keep? 
