Question

1. Express in the language of the FOL the following mathematical statement:

If S is a set of elements and * is a binary operation in S for which the following four assumptions hold, where = is an equivalence relation:

a. S is closed under * ( if x and y are in S then x * y is also in S)

b. * is associative (x, y, and z in S (x * y) * z = x * (y * z) )

c. S has an identity ( e S a S a * e = e * a)

d. every element in S has an inverse ( a S a-1 S (a * a-1 = e) )

then the following three statements hold:

i) x and a in S ( (x * a = y * a) x = y), i.e., every equation has unique right solution

ii) a in S the inverse of the inverse of a is a

iii) a and b in S ( (a * b))-1 = a-1 * b-1 )

A set satisfying a, b, c, and d above is referred to as a group.