a) For the first property, closure, for all positive integers a and b in the group, the element of a-b should be also in the group. However, if a is smaller than b, the value of a-b is negative and doesn’t satisfy the property. On the other hand, for all integers a and b, the element of a+b is always an integer which satisfies the property. For the second property, associativity, for all positive integers, the element of (a - b) - c should equal to a - (b - c). However, if a=1, b=3, and c=7, (a-b)-c =(1-3)-7=-9 while a-(b-c) =1-(3-7)=5 which doesn’t qualify the property. But for all integers, (a + b) + c = a + (b + c) definitely works. For the third property, identity, for all positive integers a and e, a - e = e - a = a should work. To satisfy it, e should be 0, but since it is not a positive integer, the property is not fulfilled. But for all integers, since 0 is included in an integer group, a + e = e + a = a works. For the fourth property, inverses, for all positive integers a and a’, a - a’ = a’ - a = e should work. However, since the third property is not satisfied, there is no e, so the fourth one can’t be worked. For all integers, however, there is the value of e and a + a’ = a’ + a =0 works.
b) For the first property, closure, for all positive rational numbers a and b in the group, the element of a/b should be also in the group. However, if a = 1 and b = 3, the value of a/b is not a rational number anymore, so it doesn’t satisfy the property. On the other hand, for all positive rational numbers a and b, the element of a*b is always a positive rational numbers which satisfies the property. For the second property, associativity, for all positive rational numbers, the element of (a/b)/c should equal to a/(b/c). However, if a = 1, b = 2, and c = 10, (a/b)/c = (1/2)/10 = 0.05 while a/(b/c) = 1/(2/10) =5 which doesn’t qualify the property. But for all positive rational numbers, (a*b)*c = a*(b*c) definitely works. For the third property, identity, for all positive rational numbers a and e, a/e = e/a = a should work. However, there is no way to satisfy it. But for all positive rational numbers, a*e = e*a = a works when e=1. For the fourth property, inverses, both multiplication and division work. For division, w hen a equals to a’ and for multiplication, when a equals to 1/a’.
c) For the first property, a*b works for all positive integers. For the second property, (a*b)*c = a*(b*c) also works for all positive integers. For the third property, identity, for all positive integers a and e, a*e = e*a = a works when e=1. However, for the fourth property, inverses, for all positive integers a and a’, a*a’ = a’*a =1 should work. It is satisfied when a equals to 1/a’ , but then, 1/a’ is not a positive integer anymore. So the set of positive integers with the operation of multiplication doesn’t form a group.
d) For the first property, a*b works for all positive even integers. For the second property, (a*b)*c = a*(b*c) also works for all positive even integers. For the third property, identity, for all positive even integers a and e, a*e = e*a = a works when e=1, but since 1 is not a positive even integer, it doesn’t work. Since the third property doesn’t work, the fourth one doesn’t work as well. So the set of positive even integers with the operation of multiplication doesn’t form a group.
e) For all integers, m*n = m+n+1 works. To get the identity element, m*e = e*m = m+e+1=m and e = -1. For the inverse element, m*m’ = m’*m = e which is 1. So m+m’+1 should equal to 1 and m’=-m. In brief, the identity element is -1 and the inverse of the element m is -m.
f) For all integers, m*n = m+(-1)mn works. For the identity element, m*e = e*m = m+(-1)me = m and e = 0. For the inverse element, m*m’ = m’*m = e which is 0. So m+(-1)mm’ should equal to 0. When m is an odd integer, m’=m while m is an even integer, m’=-m. In brief, the identity element is 0 and when m is an odd integer, the inverse element of element m is m, while it is an even integer, that of element m is -m.
g) For all real numbers excluding the number -1, x*y=xy+x+y works. For the identity element, x*e=e*x=xe+x+e=x. So xe+e=0. To satisfy e(x+1)=0 for all real number x, e should be 0. For the inverse element of element x, x*x’=x’*x=e which is 0. So xx’+x+x’=0 and x’(x+1)=-x. As a result, x’=(-x)/(x+1). In brief, the identity element is 0 and the inverse element of element x is (-x)/(x+1).