properties of relations I'm trying to do some chapter problems on equivalence relations. I'm stuck in the second section "properties of relations."
Question: Let $A=\{a,b,c,d\}$. Give an example of a relation $R$ on $A$ that is neither reflexive, symmetric ,or transitive.
What I tried doing was writing out all the pairs and then canceling out the ones that matched with the laws.  I was left with $\{(d,b) (d,c)\}$. Does that mean $R=\{(d,b) (d,c)\}$ is not reflexive, transitive, or symmetric ?
 A: Well, here's a way to do it.  Maybe over kill.
Let $S = A \times A$.  That is the universal relationship.
As $(x,x) \in S$ for all $x$ it is reflexive.  Let's destroy that be removing $(a,a)$.
$T = A \times A \setminus \{(a,a)\}$.
$T$ is not reflexive because $a \not T a$.
But if $x T y$ then $y T x$.  So T is symmetric.  Let's kill that.  Let's remove $(a,b)$.
$W = A \times A \setminus \{(a,a),(a,b)\}$ 
Not $W$ is not symetric because $b W a$ but $a \not W b$.
Is $W$ transitive?  To be honest, I'm not sure.  But $a R c$ and $c W d$ and $a W d$.  Let's remove $(a,d)$.
Let $R =  A \times A \setminus \{(a,a),(a,b),(a,d)\}$
$R$ is not transitive because $a R c$ and $c R d$ but $a \not R d$.  
And it's not symmetric as $a \not R a$ nor reflexive as $b R a$ but $a \not R b$.

Or we could have built from scratch;
$S = \{(a,b)\}$
Not reflexive: $a \not S a$.
Not symmetric: $a S b$ but $b \not S a$.
Transitive?  Vacuously so.  There are no $x S y$ and $y S z$.  So for all zero of those it is vacuously true $x S y; y S z \implies x S z$.
But let's kill this by adding $R = \{(a,b),(b,c)\}$
Then we have $a R b$ and $b R c$ but $a \not R c$ so it is not transitive.
A: 
Question: Let $A=\{a,b,c,d\}$. Give an example of a relation $R$ on $A$ that is neither reflexive, symmetric ,or transitive.

I built that from knowing to exclude reflexivity; there's no symmetry, and no transitivity.
Let $$R =\{(a,b ), (b, c), (c, d)\}$$
A: Well, the relation you give is indeed neither reflexive nor transitive, but there are many ways to get such a relation.  "Cancelling out pairs that don't match the laws" is not a well defined procedure.  (Well, it is defined for reflexivity, but it does not do what you want:  You would be left with $\{(a,a),(b,b),(c,c),(d,d)\}$ which is reflexive). 
I think the problem would like you to find a relation that involves all four of the members of the set.  You could, for example, add in $(a,b)$.  Or even $(a,a)$; just one element being equivalent to itself does not make the relation reflexive if others are not.
