Consider the binary operations * : R × R → R and o : R × R → R defined as ab = |a − b| and a o b = a for all a, b ∈ R. Show that '' is commutative but not associative, 'o' is associative but not commutative.

Two numbers are said to satisfy the commutative property if pq = qp. They satisfy the associative property if p*(qr) = (pq)r.
ab = |a − b|, ba = |b − a| = |a − b|
(ab)c = |a − b|c = ||a − b| − c| .. (1)
a(bc) = a*|b − c| = |a − |b − c|| .. (2)
The expressions (1) and (2) might not be the same. ⇒ '*' is commutative but not associative.
a o b = a, b o a = b
(a o b) o c = a o c = a, a o (b o c) = a o b = a
⇒ 'o' is associative but not commutative.