Exercise 12.1

Exercise . . (i) Yes, * is binary on  (ii) Yes, * is binary on A (iii) No, * is not binary on  . No, * is not binary on  . - . Yes, usual multiplication is binary on A . (i) The given operation * is closure and commutative but not associative on  . (ii) Identity does not exist and so inverse does not exist. . * . No. The given operation is not commutative and associative . (i) A B     (ii) A B     - - Answers (iii) A B ) ∧     (iv) A B ) ∨     . (i) It is commutative and associative. (ii) Identity and Inverse is exist.