This shows that R is
reflexive Further, (x, y) R (u, v) ⇒ xv = yu ⇒ uy = vx and hence (u, v) R (x, y) This
shows that R is symmetric Similarly, (x, y) R (u, v) and (u, v) R (a, b) ⇒ xv = yu and
ub = va ⇒ 
a
a
xv
yu
u
u
=
⇒ 
b
a
xv
yu
v
u
=
 ⇒ xb = ya and hence (x, y) R (a, b)