Chapter 5.1, Problem 6E

Exercises

Work exercise 5 using $U=\left\{a\right\}$.

Exercise5

Let $U=\{a,b\}.$

Define addition and multiplication in $P(U)$

by $C+D=C\cup D$

and $CD=C\cap D$. Decide whether $P(U)$ is a ring with respect to these operations. If it is not, state a condition in Definition 5.1a that fails to hold.

Definition 5.1a:

Suppose $R$

is a set in which a relation of equality, denoted by $=$, and operations of addition and multiplication, denoted by $+$

and $\cdot$, respectively, are defined. Then $R$

is a ring with respect to these operation if the following conditions are satisfied :

1) $R$

is closed under addition: $x\in R,y\in R\Rightarrow x+y\in R$

2) Addition in $R$ is associative: $(x+y)+z=x+(y+z)\forall x,y,z\in R$

3) $R$ contains an additive identity $0$: $x+0=0+x=x\forall x\in R$

4) $R$ contains an additive inverse: for each $x$ in $R$, there exists $-x$ in $R$ such that $x+(-x)=(-x)+x=0$.

5) Addition in $R$ is commutative: $x+y=y+x\forall x,y\in R$

6) $R$

is closed under multiplication: $x\in R,y\in R\Rightarrow x\cdot y\in R$

7) Multiplication in $R$ is associative: $(x\cdot y)\cdot z=x\cdot (y\cdot z)\forall x,y,z\in R$

8) Two distributive laws holds in $R$: $x\cdot (y+z)=x\cdot y+x\cdot z$ and $(x+y)\cdot z=x\cdot z+y\cdot z$

$\forall x,y,z\in R$