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Andrew Carter

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Boolean Algebra and Logic Circuits

A Boolean Algebra operation can be related with an electronic circuit in which the inputs and outputs corresponds to the statements of Boolean algebra. Though these circuits may be complicated, they all be constructed from three basic devices. These are the AND gate, the OR gate and the NOT gate. The basic rules for simplifying and combining logic gates are called Boolean algebra in honor of George Boole (1815 – 1864) who was a self-educated English mathematician who developed many of the fundamental ideas.

Logic gates are the essential building blocks of computer. They are physically put into operation by one to six or more transistors depending upon the application used. Gates have at least one input and only one output. Input and output values are the logical values true and false or 1 and 0.

Laws of Boolean Algebra

The basic Laws of Boolean Algebra which relate to the Commutative Law letting a change in position for addition and multiplication, the Associative Law allowing the elimination of brackets for addition and multiplication, the Distributive Law allowing the factoring of an expression.

1. Commutative Law – in this law the reverse the order of variables that are added or multiplied together without changing the truth of the expression.

(a) A + B = B + A
(b) A B = B A

2. Associative Law – this law allows the elimination of brackets from an expression and regrouping of the expression.

(a) (A + B) + C = A + (B + C)
(b) (A B) C = A (B C)

3. Distributive Law – this law allows the multiplying or factoring out of an expression.

(a) A (B + C) = A B + A C
(b) A + (B C) = (A + B) (A + C)

4. Identity Law – in this law a term OR with a 0 or AND with a 1 will always equal in that term

(a) A + A = A
(b) A A = A

5. Indempotent Law – in this law an input AND with itself or OR with itself is equal to that input.

(a). A + A = A
(b) A . A = A

6. DeMorgan’s Law – this laws are used to simplify Boolean equations to build equations only involving one sort of gate, generally only using NAND or NOR gates. It also states the same equivalence in reversed form that inverting the output of any gate gives in the same function as the opposite type of gate of AND or OR with inverted inputs.

When multiple levels of expression exist in an expression, you may only break one bar at a time, and it is normally easier to start simplification by breaking the uppermost bar first. To demonstrate, the [removed]A + (BC)’)’ and solve it using De Morgan’s Theorems.

De Morgan’s Law example:

(A (not B)) + ((not A) B) = not (((not A) + B) (A + (not B)))

Tags: boolean algebra, logic circuits

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