Boolean logic defines an abstract mathematical structure. By a Boolean algebra we mean a set B together with two binary operations +, ∙ on B (known as addition and multiplication respectively) and a unary operation ' on B (called complementary) satisfying the following axioms:
Axiom. 1. The operations are commutative; i.e.,
a + b = b .+ a, a ∙ b = b ∙ a for all a, b ϵ B
Axiom. 2. Each binary operation distributes over the other; i.e.,
a + (b ∙ c) = (a + b) ∙ (a + c)
and a ∙ (b + c) = (a ∙ b) + (a ∙ c) for all a, b, c ϵ B
Axiom. 3. B contains distinct identity elements 0 and 1 (known as zero element and unit element) with respect to the operations +, ∙ respectively; i.e.,
a + 0 = a, a ∙ 1 = a, for every a ϵ B.
Axiom. 4. For each a ϵ B, there exists an element a' ϵ B such that a + a' = 1 and a ∙ a' = 0.
(i) a' is called the complement of a. (a')' will be denoted by a'' and so on. Very often we shall write a ∙ b as ab.
(ii) The binary operations in the definition need not be written as + and
Instead, we may use other symbols such as ∪, ∩ (known as union and intersection respectively), or, ⋁, ⋀ (known as join and meet) to denote these operations.
(iii) A Boolean algebra is generally denoted by a 6tuple (B, +, ∙, ', 0, 1) or by (B, +, ∙, ') or, simply by the set B in it.
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1. Let A be a nonempty set and P(A) be the power set of A. Then P(A) is a Boolean algebra under the usual operations of union, intersection and complementation in P(A). The sets ∅ and A are the zero element and unit element of the Boolean algebra P(A). Observe that if A is an infinite set, then the Boolean algebra P(A) will contain infinite number of elements.
2. Let B be the set of all positive integers which are divisors of 70; i.e., B = {1, 2, 5, 7, 10, 14, 35, 70}. For any a, b ϵ B, let a + b = l.c.m of a, b; a ∙ b = h.c.f. of a, b and a' = ⁷<span style='fontsize: 50%'>/₀. Then with the help of elementary properties of l.c.m. and h.c.f. it can be easily verified that (B, +, ∙, ', 1, 70) is a Boolean algebra. Here 1 is the zero element and 70 is the unit element.
We can generalize this example with the help of the following result:
Let n > 1 be an integer and B be the set of positive integers which are divisors of n. For a, b ϵ B we define a + b = l.c.m of a, b; a ∙ b = h.c.f of a, b and a' = ⁿ/₀. Then (B, +, ∙, ', 1, n) is a Boolean algebra if and only if n is squarefree, i.e., n is not divisible by any square integer greater than 1.
Using simple properties of integers and of l.c.m. and h.c.f. we can easily show that axioms (1)(3) given in the definition of a Boolean algebra are satisfied. Now axiom (4) will hold if and only if for any a ϵ B, a and n/a have no common factor, other than 1. This condition is equivalent to n being squarefree.
If n = 50 which is not squarefree, B = {1, 2, 5, 10, 25, 50}. Observe that 5' = ⁵<span style='fontsize: 50%'>/₅ = 10 and 5 + 5' = 5 + 10 = l.c.m of 5, 10 = 10 ≠ 50. Also, 5 ∙ 5' = 5 ∙ 10 = h.c.f of 5, 10 = 5 ≠ 1. Thus {B, +, ∙, ', 1, 50} is not a Boolean algebra.
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