Given is the relation $R := \left\{(b,a), (b,b), (b,c), (c,a), (c,c)\right\}$ over the set $A := \left\{a,b,c,d\right\}$. Which pairs $(x,y) \in A \times A$ you at least need to add to $R$ such that $R$ is
(i) reflexive
(ii) antisymmetric
(iii) connex
(i) You need to add the pair $(a,a)$
(ii) A relation is antisymmetric if $\forall x,y \in A: xRy \wedge yRx \Rightarrow x=y$. But that relation is already antisymmetric, so we don't need to add any pairs.
(iii) A relation is connex, if $\forall x,y \in A: (x,y) \in R \vee (y,x) \in R $ or $x=y$.
So as it looks like, this relation is also already connex, we don't need to add any pairs.. :o
I'm a bit confused. We need to add pairs to $R$ such that it satisfies these properties (as less pairs as possible added). So we really just need to look at $R$ itself, right? Or do I also need to take care of element $d$ because it occurs in set $A$?
Did I do it correctly? If some relations are unknown to you (maybe the connex), assume that my definition is correct.
Yes, you have to care about the unused element $d$.
(i) Regarding to the definition of reflexivity of $R$, we need $\forall x\in A:\,(x,x)\in R$.
In particular, as $d\in A$, we also need to add $(d,d)$ to $R$ besides $(a,a)$ in order to obtain a reflexive relation.
(ii) Correct.
$R$ remains antisymmetric if we add more elements to the base set (with no relation).
(iii) We need to deal with $d$, it has to relate to all other elements somehow in the extended relation.
So we need to add at least $3$ pairs, but these are not uniquely determined. We might choose to put $d$ 'on the top', i.e. to add the pairs $(a,d),\ (b,d)$ and $(c,d)$.