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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

Verified 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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