`D_(8)` is a boolean algebra?

Solution:
(1) Commutative laws
The following are the tables for `+`,`*` and `'`

`+` 1 2 4 8 1  1 `1=1+1=lcm(1,1)`  2 `2=1+2=lcm(1,2)`  4 `4=1+4=lcm(1,4)`  8 `8=1+8=lcm(1,8)` 2  2 `2=2+1=lcm(2,1)`  2 `2=2+2=lcm(2,2)`  4 `4=2+4=lcm(2,4)`  8 `8=2+8=lcm(2,8)` 4  4 `4=4+1=lcm(4,1)`  4 `4=4+2=lcm(4,2)`  4 `4=4+4=lcm(4,4)`  8 `8=4+8=lcm(4,8)` 8  8 `8=8+1=lcm(8,1)`  8 `8=8+2=lcm(8,2)`  8 `8=8+4=lcm(8,4)`  8 `8=8+8=lcm(8,8)`

`*` 1 2 4 8 1  1 `1=1*1=gcd(1,1)`  1 `1=1*2=gcd(1,2)`  1 `1=1*4=gcd(1,4)`  1 `1=1*8=gcd(1,8)` 2  1 `1=2*1=gcd(2,1)`  2 `2=2*2=gcd(2,2)`  2 `2=2*4=gcd(2,4)`  2 `2=2*8=gcd(2,8)` 4  1 `1=4*1=gcd(4,1)`  2 `2=4*2=gcd(4,2)`  4 `4=4*4=gcd(4,4)`  4 `4=4*8=gcd(4,8)` 8  1 `1=8*1=gcd(8,1)`  2 `2=8*2=gcd(8,2)`  4 `4=8*4=gcd(8,4)`  8 `8=8*8=gcd(8,8)`

`'` 1 2 4 8  8 `8=1'`  4 `4=2'`  2 `2=4'`  1 `1=8'`

From the tables it is clear that `AA x, y in D_(8), x+yin D_(8) and x*yin D_(8)`

`:.` `+` and `*` are binary operations.

The symmetry about the principal diagonal of first two tables indicates the commutative laws hold.

(2) Associative laws
We verify associative laws for 1,2 and 4
`1+(2+4)=1+4=4`

`(1+2)+4=2+4=4`

and
`1*(2*4)=1*2=1`

`(1*2)*4=1*4=1`

Similarly, it may be verified for other cases.

(3) Distributive laws
We verify distributive laws for 2,4 and 8
`2+4*8=2+4=4`

`(2+4)*(2+8)=4*8=4`

and
`2*(4+8)=2*8=2`

`2*4+2*8=2+2=2`

Similarly, it may be verified for other cases.

(4) Existence of identity elements
For the zero and unit elements, we check
`1+1=1,1+2=2,1+4=4,1+8=8`

`8*1=1,8*2=2,8*4=4,8*8=8`

`:. 1` is the zero element and `8` is the unit element


(5) Existence of complement
`1+1'=1+8=8` & `1*1'=1*8=1`
`2+2'=2+4=4!=8` & `2*2'=2*4=2!=1`

`1'=8,``8'=1`
`2'!=4,``4'!=2`

`:. (D_(8),+,*,',1,8)` is not a boolean algebra.

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