3. D8 is not a boolean algebra example
`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'` |
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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.
This material is intended as a summary. Use your textbook for detail explanation.
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