Let * be a binary operation on the set Q of rational numbers as follow

1.	Let * be a binary operation on the set Q of rational numbers as follows. Find which of the binary operations are commutative and which are associative. a * b = (a – b)2
Answer» a * b = (a – b)² = (b – a)²= b * a hence * is not commutative (a * b) * c = (a – b)² * c = ((a-b)² – c)² = (a – b)⁴+ c² – 2c (a – b)² a(b c) = a* (b – c)² = [a -(b – c)]² = a² + (b – c)² – 2a (b – c)² hence not associative

Let * be a binary operation on the set Q of rational numbers as follows. Find which of the binary operations are commutative and which are associative. a * b = (a – b)2

Answer»

a * b = (a – b)² = (b – a)²= b * a

hence * is not commutative

(a * b) * c = (a – b)² * c = ((a-b)² – c)²

= (a – b)⁴+ c² – 2c (a – b)² a*(b * c)

= a* (b – c)² = [a -(b – c)]²

= a² + (b – c)² – 2a (b – c)²

hence not associative