I don't understand why this relation is not reflexive

Question

Let $S = \{a, b, c, d\}$

$R_1 : \{(a, a),(d, d)\} $
I don't understand why this relationship is not reflexive. $R_1$ is a subset of $S$x$S$ and every element in $R_1$ is related to itself. the answer key says it's not reflexive because its missing $(b,b)$ and $(c,c)$. However, the definition of reflexive is :

So I dont see why it matters if we dont have $c$ and $b$. the implication will be vacioulsy true

Answer 1

It is not reflexive because you don't have $b R b$. The definition of reflexivity asks this property for all elements of the set $S$: $a R a$, $b R b$, $c R c$, $d R d$. Here the given relation does not satisfy all these four conditions. The operator $\forall \ x$ means for all x in $S$. Not just $a$ and $d$.

Answer 2

b $\in$ S, but (b,b) $\notin S \times S$, but in the definition we need that it holds for every element in S.

Btw., if you doesn't make the prerequisite that it holds for every element in the set, you could directly conclude that every symmetric and transitive relation is also reflexive. Since a ~ b => b ~ a (symmetric) => a ~ a (transitive).

Answer 3

True, $R_1$ should be a subset of $S\times S$, but also a reflexive relation is defined to be the set of ALL pairs $(x,x)$ in $S\times S$

for the case of $S=\{a,b,c,d\}$, $R_1$ to be reflexive, $R_1:\{(a,a),(b,b),(c,c),(d,d)\} = \{(x,x):x\in S\}$

https://mathworld.wolfram.com/Reflexive.html