You are essentially asking yourself:

a,b,c

1) reflexive does(a,a) ?

2) symmetric, if (a,b) , is (b,a) ?

3) transitive, if (a,b) , is (b,a) ?

4) irreflexive, does (a,a) ?

5) antisymmetric, if (a,b) , and (b,a) then is a=b?

A)

R = {(a,b) l a squared = b squared} over the real numbers

"Equal" is a reflexive, symmetric, transitive relation

Try all the 5 conditions out for R = {(x,y) l x = y}

B)

R = {(x,y) l x divides y} over the positive integers

Note: I will use a|b for "a divides b"

Start checking:

1) a|a

2) Not symmetric , See 5)

3)if a|b and b|c then b = au and c = bv. Thus c = bv = (au)v = a(uv) and therefore a|c.

4) Since 1) holds, 4) cant

5) if a|b and b|a then .But since a,b , a=b.