What is Boolean Algebra?
Boolean algebra is a brunch of mathematical operations on binary variables (0's or 1's) 0 means false, and 1 means true. The boolean algebra was developed by George Boole and it is an important basis in digital logic design and computer science.
Basic Concepts of Boolean Algebra
Boolean algebra deals with binary numbers (0,1) and elementary algebra deals with numerical operations. Boolean algebra involves variables and logical operations, which are fundamental to its structure. The primary logical operations are AND, OR, and NOT(see the last post here).
1- AND (Conjunction): The AND operation, denoted by a dot (·) or simply by juxtaposition, outputs true only if both operands are true. For example, A⋅B, or AB is true if both A and B are true.
2- OR (Disjunction): The OR operation, denoted by a plus sign (+), outputs true if at least one operand is true. For instance, A+B is true if A, B, or both are true.
3- NOT (Negation): The NOT operation, indicated by an overline or a prime ('), inverts the value of a variable. If A is true it is out A' and vice versa.
Boolean Algebra Laws and Theorems
Boolean algebra is controlled by several fundamental laws and theorems. The basic thermoses and laws of boolean algebra:
1- commutative Law:
A . B = B . A
A + B = B + A
2- Associative Law:
A . (B . C) = (A . B) . C
3- Distribution Law:
A . (B + C) = (A . B) + (A . C)
A + (B . C) = (A + B) . (A + C)
4- Identity Law:
A + 0 = A
A ⋅ 1 = A
5- Idempotent Law:
A + A = A
A ⋅ A = A
6- Complement Law:
A + A'= 1
A ⋅ A'= 0
7- Double Negation Law:
A'' = A
The Theorems
De Morgan's Theorems
De Morgan's Theorems are powerful tools in Boolean algebra and digital logic design. They provide a straightforward method for transforming and simplifying expressions, which is essential for creating efficient digital circuits.
De Morgan's Theorems state the following:
1- First Theorem:
A . B = A + B
This theorem states that the complement of a product (AND operation) equals the sum (OR operation) of the complements.
2- Second Theorem:
A + B = A ⋅ B
This theorem states that the complement of a sum (OR operation) is equal to the product (AND operation) of the complements.
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