1. 6.4 Finite Boolean Algebra

2. A Boolean Algebra as 2n elements an is noted as Bn.
If a poset has 6 elements, it is not a Boolean Algebra because it cannot be expressed as 2n. There is no n that would satisfy 2n = 6.

3. {2,3,5}
{2,3} {2,5} {3,5} {2,3,5} {2,3} {3,5}
{2,5} {2} {3} {5}
{2} {3} {5} { }
0 0 0
{ }
111
000

4. A Boolean Algebra has an even number of elements
111
000
8 elements 23
The length of the binary element is B3

5. 20 All B0 look the same(1 element)
21 All B1 look the same (2 elements) 11
22 All B2 look the same (4 elements) 00
23 All B3 look the same (8 elements)
000
If you have an 8 element diagram that doesn’t look like this then, it is not a Boolean Algebra.

6. Note
B1 diagrams use binary length 1
B2 diagrams use binary length 2
B3 diagrams use binary length 3
Isomorphism: An isomorphism between two mathematical structures A and B is a one to one correspondence between A and B that preserves all the properties of the operations of the mathematical structure.

7. When you see the notation D30 it means all possible divisors of 30.
1 divides all numbers
2 divides 10, 6, 30
3 divides 6,15,
5 divides 10,15,30
6 divides
10 divides 30
15 divides
30 divides 30 but in Hasse diagrams we don’t show this relation.
Is this diagram a boolean algebra? Yes…it is isomorphic to the drawing on page 244.

8. We will not be learning about Boolean Algebra compliments
We will not be learning about Boolean Algebra compliments. Homework problem #27 has a question about compliments. Don’t concern yourself with this. Determine Bn ….what is n? How many elements does it have? Is the shape isomorphic with the diagram shown in the section? Is the matrix the same as the matrix for the diagram shown for a Bn ?