question_answer 44)

Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, give justification for this. (i) On Z+, define * by a * b = a – b (ii) On Z+, defined * by a * b = ab (iii) On R, define * by a * b = ab2 (iv) On Z+, define * by a * b = |a – b| (v) On Z+, define * by a * b = a  

Answer:

(i) Given a * b = a ?  Here a ? b may or may not       For example,       Let 5 ? 7 = ?2        is not a binary operation on Z+. (ii)   Given a * b =       Since for every a, , there exists a unique element  therefore & is a binary operation on Z+. (iii)  Given a * b =       Here                   Hence * is a binary operation on R. (iv)  Given a * b = |a ? b|       Here             is a binary operation on Z+. (v)  Given a * b           Here        is a binary operation on Z+.