Part B:
Example 11.57 very quickly shows that the set of positive integer divisors of 30 forms a Boolean algebra. It actually takes quite some (easy but tedious) work to verify that it satisfies all of the laws, and this is a subtle process.
Take the set S={0,1,a,b,c,d} with the complements defined as:
/bar (0)=1(,)/(b)ar (b)=d,ar (a) =c (and /bar (ar{x} )=x for any x to get the other 3 complements)
Define + via: a+b=b,c+d=c,a+c=a+d=b+c=b+d=1, and x+y=y+x,0+x=x,x+x=x, for any x,y to get the others
Define * via: a*b=a,c*d=d,a*c=a*d=b*c=b*d=c*d=0, and x*y=y*x,1*x=x,x*x=x, for any x,y to get the others
Show that this is not a boolean algebra, by pointing out which part of Definition 11.56 fails.
Part B: Example 11.57 very quickly shows that the set of positive integer divisors of 30 forms a Boolean laws,and this is a subtle process. algebra. It actually takes quite some (easy but tedious) work to verify that it satisfies all of the Take the set S = {0,1,a,b,c,d} with the complements defined as: 0 = 1, b = d, a = c (and x = x for any x to get the other 3 complements) Define + via: a+b = b, c+d=c, a+c=a+d=b+c=b+d=1, and x+y=y+x, 0+x = x, x+x = x, for any x,y to get the others Define - via: a-b = a, c-d=d, ac=a-d=b-c=b-d=c-d=0, and x:y=yx, 1-x = x, xx = x, for any x,y to get the others
Show that this is not a boolean algebra, by pointing out which part of Definition 11.56 fails.