Determine whether ababc where c is at most ab and abba are c

Determine whether a*b=a^b+c (where c is at most a^b) and a*b=b^a are commutative and/or associative.

Determine whether the definition of * gives a binary operation on the set. If * gives a binary operation on the set, then determine whether * is commutative and whether * is associative. the operation * defined on Z^+ by a * b = a^b + c, where c is at most a^b the operation * defined on Z^+ by a * b = b^a

Solution

commutative law is as follows

example + operator

2+3=3+2=5

and associative law can be proved by

2+(3+4)=(2+3) +4 = 9

1) a*b=a^b+c

now b*a= b^a + c which is not equal to a*b

Hence not follows commutative law

a*(b*c) =a*(b^c+k)= a^{b^c+k}+g

(a*b)*c=(a^b+d)*c=((a^b+d)^c + k

Hence not follows associative law

2) a*b=b^a and b*a=a^b hence not follows commutative law

(a*b)*c=b^a*c= c^{b^a}

a*(b*c)= a*c^b=(c^b)^a=c^ba Hence not follows associative law