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State whether the following statements are true or false. Justify.
(i) For an arbitrary binary operation * on a set N,  a * a = a ,  for all a * N.
(ii) If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a



Answer :

crisforp
(i) false;
Let be N = { 0, 1 , 2}; x * y = x + y => a * a = 2a ∈ { 0, 2 , 4}, which is not a( just for a = 0);

(ii) true;
b * c = c * b => a * ( b * c ) = a * ( c * b );                  (1)
                   but, a * ( c * b ) = ( c *  b ) * a;               (2)

(1) and (2) => a * (b * c) = (c * b) * a.

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