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 + ab
Answer» a * b = a + ab b * a = b + ab ∴ a * b ≠ b * a, hence * is not commutative (a * b) * c = (a + ab) * c = (a + ab) + (a + ab) c = a + ab + ac + abc a* (b * c) = a * (b + bc) = a + a (b + bc) = a + ab + bca ≠ (a * b) * c hence * is 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 + ab

Answer»

a * b = a + ab

b * a = b + ab

∴ a * b ≠ b * a,

hence * is not commutative

(a * b) * c = (a + ab) * c

= (a + ab) + (a + ab) c

= a + ab + ac + abc a* (b * c)

= a * (b + bc) = a + a (b + bc)

= a + ab + bca ≠ (a * b) * c

hence * is not associative