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Morgan's Theorem

Morgan's Theorem allows for the transformation of product functions into sum functions and vice versa, which is useful for developing digital circuits using a single type of logic gate. The theorem states that the complement of a product of variables is equal to the sum of the complements of the variables, and that the complement of a sum is equal to the product of the complements. This means that OR functions can be implemented as AND with inverted inputs, and vice versa.

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9 views6 pages

Morgan's Theorem

Morgan's Theorem allows for the transformation of product functions into sum functions and vice versa, which is useful for developing digital circuits using a single type of logic gate. The theorem states that the complement of a product of variables is equal to the sum of the complements of the variables, and that the complement of a sum is equal to the product of the complements. This means that OR functions can be implemented as AND with inverted inputs, and vice versa.

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MORGAN'S THEOREM

Application. Morgan's Theorem allows


transform product functions into sum functions
vice versa. Its main practical application is
to carry outdigital circuits using a single type of
gate. It is also widely used in thealgebra
Booleanto obtain the complement of a
expression or a function, also to simplify
expressions andboolean functions.
Morgan's theorem is a very useful tool.
to develop digital circuits, which allows us
obtain the function of alogic gate with the
combination of other logical gates, by
example can perform the function of the gate
NAND with an OR gate and two gates
inverters, and one can obtain the function of a
NOR gate with an AND gate and
twogates
investors.

Example of practical application


In this example, we will obtain the function of a three-input NAND gate.
inputs from the combination of a three-input OR gate and
three inverter gates, or the combination of three OR gates of two
inputs and three inverter gates.
NAND gate

Combination of the OR gate and the three inverters

In this example we will obtain the function of a three-input NOR gate.


outputs from the combination of a three-input AND gate and
three inverter gates, or the combination of three AND gates of two
inputs and three inverter gates.
NOR gate
Combination of the AND gate and the three inverters

Morgan's theorem is used to transform


functions that ADD in functions that
MULTIPLY or VICE VERSA
The application of this theorem is fundamental because it allows
replace an OR gate with an AND gate or create a circuit
logical USING ONLY NAND gates.

Now we represent the original function with gates


combined...
We will see how the function obtained after is represented.
apply Morgan's theorem...

The advantage of this practice is that we only have to buy a


only type of integrated (NAND GATES)

Theorems Of Morgan proposed two theorems that


they constitute a very important part of Boolean algebra.
These theorems demonstrate the equivalence between:
NAND gates and Negative-OR - NOR gates and
Negative-AND First Theorem of First Theorem of De
Morgan
The complement of a product of variables is equal to the
sum of the complements of the variables. In this way
equivalent:
The complement of two or more variables to which it is applied
the AND operation is equivalent to applying the OR operation
to the complements of each variable. Formula for
XY = X + Y for two variables.
Equivalent gate and truth table: Second Theorem of
Second De Morgan's Theorem
The complement of a sum of variables is equal to
product of the complements of the variables. • In form
equivalent:
The complement of two or more variables to which it is applied
the OR operation is equivalent to applying the AND operation
to the complements of each variable.
Formula to express the theorem for two variables: X + Y
= X Y • Equivalent gate and truth table:
Morgan's Theorems for More Than Two Variables
De Morgan's Theorems also apply to
expressions in which there are more than two variables: XYZ =
X + Y + Z X + Y + Z = XYZ

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