MATH 631 Homework  September 27, 1999
Section 2: A Time for Abstraction

This assignment contains 4 problems to be turned in for a grade. The tentative due date for these problems is Friday, October 8.

For further practice on this material, do the problems scattered throughout the notes for this section.

Problem 1. Let A be set and let $R=\mathcal{P}(A)$ be the set of all subsets of A. For $B,C\in R$, define B+C to be the symmetric difference of B and C, that is $B+C=\{\,a\in A\mid a\text{ is in exactly one of }B,C\,\}$ and $B\cdot C=B\cap C$. Show that R becomes a ring with these operations.

Problem 2. A ring R is called a Boolean ring if it has the property that a*a=a for all $a\in R$.

1.

Show that the ring R in Problem 1 above is a Boolean ring.

2.

Show that if R is a Boolean ring, then a+a=0 for all $a\in A$.

3.

Show that every Boolean ring is commutative.

Problem 3. Let (S,*) be a semigroup with identity e and let G be the set of all elements of S that are invertible (have an inverse in S with respect to *). Show that (G,*) is a group.

Problem 4. Let (S,*) be a semigroup, let $x,y\in S$, and suppose that x*y=y*x.

1.

Prove that for every positive integer n, we have (x*y)ⁿ=xⁿ*yⁿ.

2.

Suppose that S is a group. Prove that x*y^-1=y^-1*x, x^-1*y=y*x^-1, and x^-1*y^-1=y^-1*x^-1. (Don't do unnecessary proofs.)

3.

Again, suppose that S is a group. Use Parts (1) and (2) to deduce that (x*y)ⁿ=xⁿ*yⁿ holds for all integers n.