prove that |ab|=|a||b| Solution method-1: exhaustive proof |ab| given a/b is +/+ : (a*b) -/+ : -(a*b) +/- : -(a*b) -/- : (a*b) |a||b| given a/b is +/+ : (a)*(b) -/+ : (-a)*(b) +/-: (a)*(-b) -/-: (-a)*(-b) use axioms to prove equality for each of the 4 cases method 2: If a = 0, or b = 0, then |ab| = |a||b| = 0 is true. If not, then look at four cases: i) a>0, b>0 |ab| = ab = |a||b| ii) a>0, b<0 |ab| = -ab = a(-b) = |a||b|, since a= |a| and -b = |b| iii) a<0, b>0 |ab| = -ab = (-a)(b) = |a||b| iv) a<0, b<0 |ab| = ab = (-a)(-b) = |a||b|.