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Problem: Is Boolean Algebra D8 [ Calculator, Method and examples ]

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Your problem -> Is Boolean Algebra D8

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

 + 1 2 4 8 1 1 2 4 8 2 2 2 4 8 4 4 4 4 8 8 8 8 8 8
 * 1 2 4 8 1 1 1 1 1 2 1 2 2 2 4 1 2 4 4 8 1 2 4 8
 ' 1 2 4 8 8 4 2 1

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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