x*y=x-yLet the binary operation * be defined on Z by the given rule. Determine whether Z is a group with respect to and whether it is an abelian group. State which, if any, conditions fail to hold.
You should check the four axioms of groups such that:
G1: closure: For `x, y in Z` , then `x * y in Z`
Notice that the difference of two integer numbers yields an integer number also, hence, the first axiom G1 holds.
You need to look for an element e such that:
`x*e = e*x = x`
You need to substitute `x - e` for `x*e ` such that:
`x - e = x => -e = x - x => e = 0`
Hence, since there exists an identity element e = 0, then the second axiom G2 holds.
You need to look for an element such that:
`x*x^-1 = x^(-1)*x = e`
`x - x^(-1) = 0 => x^(-1) = x`
Hence, since there exists an inverse element `x^(-1) = x` , then the third axiom G3 holds.
You need to check if the identity `(x*y)*z = x*(y*z)` holds such that:
`(x-y)*z = x*(y-z) => x - y - z = x - (y- z) => x - y - z = x - y + z`
Since the fourth axiom does not hold, hence, `(Z,*)` does not represent a group.