Answer Happy

Determine the complement of each element B belongs to
Example 13.3.4. Is this lattice a Boolean Algebra? Why? PLEASE SHOW
CLEAR STEP BY STEP. THANKS!
Example 13.3.4: Set Complement is a
Complement. In Chapter 1, we defined the complement of a subset of
any universe. This turns out to be a concrete example of the
general concept we have just defined, but we will reason through
why this is the case here. Let L = P(A), where A =
{a,b,c}. Then [L; ∪, ∩] is a bounded lattice with
0 = ∅ and 1 = A. To find the complement, if it exists, of B = {a,
b} ∈ L, for example, we want D such that
{a, b} ∩ D = ∅ and
{a, b} ∪ D = A
It’s not too difficult to see that D = {c}, since we need to
include c to make the first condition true and can’t include a or b
if the second condition is to be true. Of course this is precisely
how we defined Ac in Chapter 1. Since it can be shown that each
element of L has a complement (see Exercise 1), [L; ∪, ∩] is a
complemented lattice. Note that if A is any set and L = P (A), then
[L; ∪, ∩] is a complemented lattice where the complement of B ∈ L
is Bc = A − B. □