A Latin square is is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly

[answerhappygod]

A Latin square is is an n × n array filled with n
different symbols, each occurring exactly once in each row and
exactly once in each column.
Note that the Cayley table on page 31 is a Latin square. Use
this fact to show:
1) For every x in D4, there exists y in D4 such that
xy=R0.
2) For each of the eight elements x of D4, find the
corresponding y such that xy=R0.
3) For each of the eight pairs x,y from part 2 (where xy=R0),
show that yx=R0. Conclude that for all x,y in D4, if xy=R0, then
xy=yx.
4) For every x,y in D4, xy is also in D4.
5) Show (HV)R90=H(VR90).
6) Show that xR0=R0 x =x for all x in D4.
7) Show that (xy)z=x(yz) for all x,y,z in D4 [Hint: Use the fact
that function composition is always associative.]
8) Find a example of x,y in D4 such that xy and yx are NOT
equal.