Assignment:

1.  Reassociate  (((ab)c)d)e to  a(b(c(de))) using only 

the associative law.


(((ab)c)d)e = ((ab)c)(de) = (ab)(c(de)) = a(b(c(de)))


            2
2.  Solve  x  + x  + 1 = 0   in Z    and Z     and Z
                                 7        11        19

Modulo 7 the roots are  2,4   (x-2)(x-4)

Modulo 11 there are no roots

Modulo 19 the roots are 7,11   (x-7)(x-11)

3.  Define a product on the Rationals by

aob = a + b + ab.

(a)  Show the product is commutative.

aob = a+b+ab = b+a+ba = boa

(b)  Show the product has an identity.  (what is the identity

element).

0ob = 0+b+0b = b

(c)  Show that the product is associative. 

(aob)oc = (a+b+ab)oc = (a+b+ab) + c + (a+b+ab)c

        = a+b+c+ab+ac+bc+abc

ao(boc) = ao(b+c+bc) = a+(b+c+bc)+a(b+c+bc)

        = a+b+c+ab+ac+bc+abc

Since the two are identical, the product is associative.


(d)  Show that every element except one of them has an inverse.  

Which element does not have the inverse? 


aox = 0    <==> a+x+ax = 0 <==> x(1+a) = -a <==>  x = -a/(1+a)

Every element has an inverse except -1

and (-1)ox = -1+x+(-x) = -1  So there is no x such that (-1)ox = 1.