In this article, we discusses the theorems that are often used in digital systems, especially in simplifying circuit equations. There are 17 formulas that can be used in simplifying logical equations, 15 formulas known as Boolean formulas and 2 formulas known as De Morgan’s theorem.
Boolean theorem with a number of formulas is very helpful in designing a logic circuit, especially in simplifying logical equations.
Below are given 17 Boolean formulas.
 X * 0 = 0
 X * 1 = X
 X + 0 = X
 X + 1 = 1
 X * X = X
 X + X = X
 X * X’ = 0
 X + X’ = 1
 x + y = y + x (Commutativ law)
 x * y = y * x ( Kommutativ law)
 x+ (y+z) = (x+y) +z = x+y+z (associativ law)  x (yz) = (xy) z = xyz (associativ law)
[13a ] x (y+z) = xy + xz (Distributive Law)
[13b] (w+x)(y+z) = wy + xy + wz + xz (Distributive Law)
 x + xy = x (Distributive Law)  x + xy = x (Law Absorption)
 x + x’y = x + y (Law of implementation)
x + xy = x (1+y)
= x * 1 
= x 
De Morgan Theorem is very useful for simplifying logical equations. The two theories are:
 (x+y)’ = x’ * y’
 (x*y)’ = x’ + y’
Theorem  states that if the OR output is inverted, the result is will be the same if each input variable is reversed and then ANDed. Theorem  states that if the AND output is inverted, the result will be the same if each input variable is inverted and then to be OR.
X = [(A’+C) * (B+D’)]’
= (A’+C)’ + (B+D’)’ 
= (A”*C’) + (B ‘+D”) 
= AC’ + B’D
From the formulas  and , it can be expanded with 3 inputs and we get the following formula:
 (x+y+z)’ = x’ * y’ * z’
 (xyz)’ = x’ + y’ + z’
Implications of the DeMorgan Theorem
Formula  : (x+y)’ = x’ * y’
Figure 1 Implications of OR to AND
Formula  : (x*y)’ = x’ + y’
Figure 2 Implications of the AND to OR
Relationship between NAND & NOR Gates
NAND gates are very broad and flexible in application, one of the advantages is that they can used to form other gate functions (OR, AND, and NOT) in a logic circuit.
NOT Gate from NAND
AND gate from NAND
OR Gate from NAND
Figure 3 NOT, AND and OR gates from NAND gate
In the same way as above, NOR gates can be assembled to produce other gate functions in a Boolean operation.
NOT Gate from NOR
OR Gate from NOR
AND Gate from NOR
Figure 4 NOT, AND and OR gates of NOR gate
Representation of Alternative Symbol
Alternative Symbols for each gate are obtained from standard symbols by performing the steps below.
- Invert all standard input and output symbols by giving a small circle for those who do not have a small circle, and removing a small circle if one is already installed.
- Change the operation symbol from AND to OR, or from OR to AND. (only for INVERTER, the operation symbol does not change)For example:
OR Gate: (previously without a small circle, and the symbol changes)
Image of an alternative OR symbol
NOR Gate: (before there was a small circle, and the symbol changed)
Image of an alternative NOR symbol
NOT Gate: (symbol does not change)
Thus, the same process for AND and NAND gates.
Logic gates can be used in electronic circuits