Is this reflexive, symmetric, antisymmetric or transitive? Set A contains all points $(x, y)$ on a coordinate plane. The relation $R$ is defined as: point $(x_1,y_1)$ is related to point $(x_2,y_2)$, if $y_1=y_2$. 
Is this set (A) reflexive, symmetric, antisymmetric or transitive? 
How would I go about solving this?
Thanks,
 A: $R$ is reflexive if $zRz$ for all $z$ in the set $A$. 
Since for any point $(x,y)$ in $A$, $y=y$, then $(x,y)R(x,y)$ and $R$ is reflexive.
$R$ is symmetric if whenever $z_1 R z_2$ then $z_2Rz_1$ also holds. Since equality itself is symmetric, if $$(x_1, y_1) R (x_2, y_2)$$ (which means $y_1 = y_2$), then we can say that $$(x_2, y_2) R (x_1, y_1)$$
$R$ is antisymmetric if whenever $z_1 R z_2$ and $z_1 \neq z_2$, then $z_2Rz_1$ never holds.
Note that it is relatively hard for $R$ to be both symmetric and antisymmetric (outside of degenerative cases). 
$R$ is transitive if $z_1Rz_2$ and $z_2Rz_3$ implies $z_1Rz_3$. I will let you fill in the details on this.
A: Since $(x,y)R(x,y) \iff y=y$ the relation is reflexive
$(x,y)R(a,b) \implies y=b \implies b=y \implies (a,b)R(x,y)$, hence the relation is symmetric
$(x,y)R(a,b)$ and $(a,b)R(c,d) \implies y=b=d \implies y=d \implies (x,y)R(c,d)$, so the relation is transitive
Note that the relation isn't antisymmetric, as $(x_1,y_1)R(x_2,y_2) \iff y_1=y_2$, but not necessarily that $x_1 = x_2$.
