question_answer 68)

Consider the binary operations * : R× defined as a * b = |a ? b| and a o b = a,  Show that * is commutative but not associative, o is associative but not commutative. Further, showthat  [If it is so, we say that the operation * distributes over the operation o.] Does o distribute over * ? Justify your answer.  

Answer:

Given * R × R  R defined as a * b = |a ? b|       Here a * b = |a ? b| = |1 ? (b ? a)|             = |b ? a| = b * a        is commutative.       Also (a * b) * c = |a ? c| * c = ||a ? b| ? c|       and a * (b * c) = a *|b ? c| = |a ? |b ? c||       Clearly, (a * b) *                  is not associative but commutative.       Also given o : R × R  R defined as aob = here aob = a and boa = b        is not commutative.. Also, (aob) oc = aoc = a       and ao (boc) = aob = a        (aob) oc = ao (boc)       ?o? is associative but not commutative       Further a * (boc) = a * b = |a ? b|       Also (a * b) o (a * b) = (|a ? b|) o (|a ? b|)             = |a ? b|               operation ?x? distributes over the operation ?0?       Also ao (b * c) = ao (|b ? c|) = a       and (a o b) * (b o c) = a * a = |a ? a| = 0             operation o is not distributive over *.