A+B [ AC + ( B+C')D] by reducing method
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Boolean Algebra and Reduction Techniques
Boolean Algebra Laws and Rules
There are three laws of Boolean Algebra that are the same as ordinary algebra.
The Commutative Law
addition A + B = B + A (In terms of the result, the order in which variables are ORed makes no difference.)
multiplication AB = BA (In terms of the result, the order in which variables are ANDed makes no difference.)
The Associative Law
addition A + (B + C) = (A + B) + C (When ORing more than two variables, the result is the same regardless of the grouping of the variables.)
multiplication A(BC) = (AB)C (When ANDing more than two variables, the result is the same regardless of the grouping of the variables.)
The Distributive Law - The distributive law is the factoring law. A common variable can be factored from an expression just as in ordinary algebra.
A(B + C) = AB + AC
(A + B)(C + D) = AC + AD + BC + BD Remeber FOIL(First, Outer, Inner, Last)?
Ten Basic Rules of Boolean Algebra
Anything ANDed with a 0 is equal to 0. A * 0 = 0
Anything ANDed with a 1 is equal to itself. A * 1 = A
Anything ORed with a 0 is equal to itself. A + 0 = A
Anything ORed with a 1 is equal to 1. A + 1 = 1
Anything ANDed with itself is equal to itself. A * A = A
Anything ORed with itself is equal to itself. A + A = A
Anything ANDed with its own complement equals 0. 2.png
Anything ORed with its own complement equals 1. 1.png
Anything complemented twice is equal to the original. 4.png
The two variable rule. 3.png
Simplification of Combinational Logic Circuits Using Boolean Algebra
Complex combinational logic circuits must be reduced without changing the function of the circuit.
Reduction of a logic circuit means the same logic function with fewer gates and/or inputs.
The first step to reducing a logic circuit is to write the Boolean Equation for the logic function.
The next step is to apply as many rules and laws as possible in order to decrease the number of terms and variables in the expression.
To apply the rules of Boolean Algebra it is often helpful to first remove any parentheses or brackets.
After removal of the parentheses, common terms or factors may be removed leaving terms that can be reduced by the rules of Boolean Algebra.
The final step is to draw the logic diagram for the reduced Boolean Expression.
Some Examples of Simplification
Perform FOIL (Firt - Outer - Inner - Last)
AA = A (Anything ANDed with itself is itself)
Find a like term (A) and pull it out. (There is an A in A, AC, and AB). Make sure you leave the BC alone at the end.
Anything ORed with a 1 is a 1 (1+C+B=1).
Anthing ANDed with a 1 is itself (A1=A)
Some Examples of Simplification (cont.)
Find like term (B) and pull it out.
Anything ORed with its own complement equals 1.
Anything ANDed with 1 is itself.
Some Examples of Simplification (cont.)
Find like term and pull them out. Make sure you leave the one.
Anything ORed with a 1 is 1.
Anything ANDed with a 1 is itself
Some Examples of Simplification (cont.)
Find like terms and pull them out.
Anything ORed with its own complement equals 1.
Anything ANDed with 1 equals itself.
NOTE: I will workout many examples in the video.
DeMorgan's Theorem
De Morgan's theorem allows large bars in a Boolean Expression to be broken up into smaller bars over individual variables.
De Morgan's theorem says that a large bar over several variables can be broken between the variables if the sign between the variables is changed.
De Morgan's theorem can be used to prove that a NAND gate is equal to an OR gate with inverted inputs.
De Morgan's theorem can be used to prove that a NOR gate is equal to an AND gate with inverted inputs.
In order to reduce expressions with large bars, the bars must first be broken up. This means that in some cases, the first step in reducing an expression is to use De Morgan's theorem.
It is highly recommended to place parentheses around terms where lines have been broken.
For example:
DeMorgan (cont.)
Applying DeMorgan's theorem and the distribution law:
Bubble Pushing
Bubble pushing is a technique to apply De Morgan's theorem directly to the logic diagram.
Change the logic gate (AND to OR and OR to AND).
Add bubbles to the inputs and outputs where there were none, and remove the original bubbles.
Logic gates can be De Morganized so that bubbles appear on inputs or outputs in order to satisfy signal conditions rather than specific logic functions. An active-low signal should be connected to a bubble on the input of a logic gate.