Boolean Logic Defines


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.


Note:

(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 6-tuple (B, +, ∙, ', 0, 1) or by (B, +, ∙, ') or, simply by the set B in it.

Examples:

1. Let A be a non-empty 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='font-size: 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:

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 square-free, i.e., n is not divisible by any square integer greater than 1.

Proof:

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 square-free.

Note.

If n = 50 which is not square-free, B = {1, 2, 5, 10, 25, 50}. Observe that 5' = ⁵<span style='font-size: 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.


1. Boolean Logic Defines.

Boolean Algebra





Boolean Algebra

From Boolean Logic Defines to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Tangrams Math | Traditional Chinese Geometrical Puzzle | Triangles

    Apr 17, 24 01:53 PM

    Tangrams
    Tangram is a traditional Chinese geometrical puzzle with 7 pieces (1 parallelogram, 1 square and 5 triangles) that can be arranged to match any particular design. In the given figure, it consists of o…

    Read More

  2. Time Duration |How to Calculate the Time Duration (in Hours & Minutes)

    Apr 17, 24 01:32 PM

    Duration of Time
    We will learn how to calculate the time duration in minutes and in hours. Time Duration (in minutes) Ron and Clara play badminton every evening. Yesterday, their game started at 5 : 15 p.m.

    Read More

  3. Worksheet on Third Grade Geometrical Shapes | Questions on Geometry

    Apr 16, 24 02:00 AM

    Worksheet on Geometrical Shapes
    Practice the math worksheet on third grade geometrical shapes. The questions will help the students to get prepared for the third grade geometry test. 1. Name the types of surfaces that you know. 2. W…

    Read More

  4. 4th Grade Mental Math on Factors and Multiples |Worksheet with Answers

    Apr 16, 24 01:15 AM

    In 4th grade mental math on factors and multiples students can practice different questions on prime numbers, properties of prime numbers, factors, properties of factors, even numbers, odd numbers, pr…

    Read More

  5. Worksheet on Factors and Multiples | Find the Missing Factors | Answer

    Apr 15, 24 11:30 PM

    Worksheet on Factors and Multiples
    Practice the questions given in the worksheet on factors and multiples. 1. Find out the even numbers. 27, 36, 48, 125, 360, 453, 518, 423, 54, 58, 917, 186, 423, 928, 358 2. Find out the odd numbers.

    Read More