Question

Let $\mathit{S}\mathbf{=}\left\{a,b,c\right\}$and $\mathit{P}\left(S\right)$be the set of all subsets of $\mathit{S}$. The elements of $\mathit{P}\left(S\right)$are as follows:

$$\begin{gathered} \mathit{S}\mathbf{=}\left\{a,b,c\right\}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{D}\mathbf{=}\left\{a,b\right\}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{E}\mathbf{=}\left\{a,c\right\}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{F}\mathbf{=}\left\{b,c\right\}\mathbf{;}\mathbf{}\mathbf{} \\ \mathit{A}\mathbf{=}\left\{a\right\}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{B}\mathbf{=}\left\{b\right\}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{C}\mathbf{=}\left\{c\right\}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{0}\mathbf{=}\mathbf{\varnothing }\mathbf{;} \end{gathered}$$

Define addition and multiplication in $\mathit{P}\left(S\right)$by these rules:

role="math" localid="1647382883066" $\mathit{M}\mathbf{+}\mathit{N}\mathbf{=}\left(M-N\right)\mathbf{\cup }\left(N-M\right)$and $\mathit{M}\mathit{N}\mathbf{=}\mathit{M}\mathbf{\cap }\mathit{N}$.

Write out the addition and multiplication tables for $\mathit{a}\mathit{b}\mathbf{=}\mathbf{0}$. Also, see Exercise 44.

Short answer

The required tables are:

+	A	0	S	B	E	D	F	C
A	0	A	F	D	C	B	S	E
0	A	0	S	B	E	D	F	C
S	F	S	0	E	B	C	A	D
B	D	B	E	0	S	A	C	F
E	C	E	B	S	0	F	D	A
D	B	D	C	A	F	0	E	S
F	S	F	A	C	D	E	0	B
C	E	A	D	F	A	S	B	0

Another table

$\times$	A	0	S	B	E	D	F	C
A	0	0	0	0	0	0	0	0
0	0	A	A	0	A	A	0	0
S	0	A	S	B	E	D	F	C
B	0	0	B	B	0	B	B	0
E	0	A	E	0	E	A	C	C
D	0	A	D	B	A	D	B	0
F	0	0	F	B	C	B	F	C
C	0	0	C	0	C	0	C	C

Step-by-step solution

Write the addition table for ab=0

It is given that the addition of elements $M,N\in P\left(S\right)$is defined as:

$M+N=\left(M-N\right)\cup \left(N-M\right)$

Form the addition table as:

+	A	0	S	B	E	D	F	C
A	0	A	F	D	C	B	S	E
0	A	0	S	B	E	D	F	C
S	F	S	0	E	B	C	A	D
B	D	B	E	0	S	A	C	F
E	C	E	B	S	0	F	D	A
D	B	D	C	A	F	0	E	S
F	S	F	A	C	D	E	0	B
C	E	A	D	F	A	S	B	0

Write the multiplication table for ab=0

It is given that the multiplication of elements $M,N\in P\left(S\right)$is defined as:$MN=M\cap N$

Form the multiplication table as:

$\times$	A	0	S	B	E	D	F	C
A	0	0	0	0	0	0	0	0
0	0	A	A	0	A	A	0	0
S	0	A	S	B	E	D	F	C
B	0	0	B	B	0	B	B	0
E	0	A	E	0	E	A	C	C
D	0	A	D	B	A	D	B	0
F	0	0	F	B	C	B	F	C
C	0	0	C	0	C	0	C	C