Any relation is reflexive if aRa for all a in the set. In other words every element is related to itself - hence 'reflexive'.
So for #1, is aRa for all a in the set? Clearly not, because a - a = 0, which is not negative.
A relation is symmetric if for any a and b in the set, aRb if and only if bRa. So is #1 symmetric? In other words, does a - b < 0 mean that b - a < 0? Again, clearly not. In fact, the opposite is true. (This means that the relation is anti-symmetric.)
A relation is transitive whenever aRb and bRc always means that aRc. So here if a - b < 0 and b - c < 0, does it follow that a - c < 0? A moment's thought will tell you that a - c = (a - b) + (b - c), and so the answer is yes, because we are adding together two negative numbers. So #1 is transitive.
(i) Is a - a even? Yes, because it's zero, of course. So it's reflexive.2. Z / aRb ⇐⇒ a−b is even
(ii) If a - b is even, is b - a even? Yes, because (b - a) = -(a - b). So it's symmetric.
(iii) If a - b is even and b - c is even, is a - c even? Yes. Again, a - c = (a - b) + (b - c). So it's transitive.
So #2 is an equivalence relation, and the integers are separated (partitioned) into just two equivalence classes: all the even numbers in one class, and all the odd numbers into the other.
(i) Reflexive? Is |a| = a, for all a in R? What about a = -3?3. R / aRb ⇐⇒ |a|=b
(ii) Symmetric? If |a| = b, does it follow that |b| = a? Again, what about a = -3 (and b = 3)?
(iii) Transitive? If |a| = b, and |b| = c, must it follow that |a| = c?
Ask the same three questions again, as we did in #3, but this time it's |a| = |b|.4. R/ aRb ⇐⇒ |a|=|b|
... and again here. Try some numbers, including as many combinations of positive and negative as you can think of.5. R/aRb ⇐⇒ |a|b=a|b|
Go carefully through #2 again, but this time with |a| and |b|. Does changing the signs (if necessary) to make it positive make any difference to whether the numbers are odd and even?6.Z/aRb⇐⇒|a| − |b| is even
You might also like to look at another posting where I give further examples of equivalence relations: http://www.mathhelpforum.com/math-he...tml#post239094