Finding the smallest binary relation in a given set including a relation The following binary relation of the set {a, b, c, d, e} is given:
R = { (a,b), (a,c), (b,c) }
What I have to do is to find the smallest reflexive / symmetric / transitive / antisymmetric relation including R.
I know what these relations are all about, but what I have trouble with is how to find them. For example, I see no reflexive relation in the set. Does "including R" mean that i can extend R with any element from the given set? So for example, if I am looking for a reflexion can I just add (a, a)?
If I can extend the set, my solution would be:
Reflexive: Add (a,a)
Symmetric: Add (b,a)
Transitive: Already satisfied
Antisymmetric: Already satisfied? I am not sure about this one

 A: You're on the right track. "Containing" means just what you would think, since $R$ is a set, we want another set of pairs of elements $(x,y)\in S\times S$ so that $R\subseteq R'$ and $R'$ has one of the properties you mention.
First you need to add $(x,x)$ for every $x\in S=\{a,b,c,d,e\}$ which will give you $R_r=R\cup\{(a,a),(b,b),(c,c), (d,d), (e,e)\}$--the smallest reflexive relation containing $R$.
For a symmetric one, you need to add all of the elements of $R$ where you switch $x$ and $y$, i.e. $R_s=R\cup\{(b,a), (c,a), (c,b)\}$
Continuing for the others, we need a transitive one, so we need to see if there are $xRy$ and $yRz$, and in-fact there is exactly one, namely since $aRb$ and $bRc$ we need to add $aRc$, but you already have $(a,c)\in R$, so $R_t=R$.
Finally anti-symmetry is already satisfied, since there are no $x,y\in S$ such that $xRy$ and $yRx$, so $R=R_a$ is already anti-symmetric. If you have trouble seeing how this is, remember that antisymmetric is parsed in logical symbols as
$$\forall x,y\in S (xRy\land yRx\implies x=y)$$
Since the set of $x,y\in S$ such that $xRy$ and $yRx$ is empty, the condition is vacuously true because all $\forall z\in\varnothing$ statements are vacuously true because the negation is $\exists z'\in\varnothing$ which is clearly false, since there are no elements in an empty set.
