(This question originates from Pinter's abstract algebra, Chapter 19 Exercise A2.)
Addition and multiplication are defined as follows.
If $A$ and $B$ are elements of $P_D$ (that is, subsets of $D$), then
\begin{align*} A+B &= (A-B) \cup (B-A) \text{ and } AB = A\cap B \end{align*}
Let $P_3$ be $P_D$ where $D=\{a,b,c\}$. Let $A=P_3, J=\{\emptyset,\{a\}\}.$
List the elements of $A/J$, and then write the addition and multiplication tables of $A/J$.
Attempt:
The elements of $A/J$ are the 4 cosets:
- $J=\{\emptyset,\{a\}\}$
- $J+\{b\}=\{\{b\},\{a,b\}\}$
- $J+\{c\}=\{\{c\},\{a,c\}\}$
- $J+\{b,c\}=\{\{b,c\},\{a,b,c\}\}$
Addition table: \begin{array}{c | c c c c } + & J & J+\{b\} & J+\{c\} & J+\{b,c\} \\ \hline J & J & J+\{b\} & J+\{c\} & J+\{b,c\} \\ J+\{b\} & J+\{b\} & J & J+\{b,c\} & J+\{c\} \\ J+\{c\} & J+\{c\} & J+\{b,c\} & J & J+\{b\} \\ J+\{b,c\} & J+\{b,c\} & J+\{c\} & J+\{b\} & J \\ \end{array}
Multiplication table: \begin{array}{c | c c c c } \times & J & J+\{b\} & J+\{c\} & J+\{b,c\} \\ \hline J & J & J & J & J \\ J+\{b\} & J & J+\{b\} & J & J+\{b\} \\ J+\{c\} & J & J & J+\{c\} & J+\{c\} \\ J+\{b,c\} & J & J+\{b\} & J+\{c\} & J+\{b,c\} \\ \end{array}
Do they look right?