## BOOLEAN ALGEBRA

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

#### 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.

Constant Theorem

[1] X * 0 = 0

[2] X * 1 = X

[3] X + 0 = X

[4] X + 1 = 1

#### Expansion Theorem

[5] X * X = X

[6] X + X = X

#### Complement Theorem

[7] X * X’ = 0

[8] X + X’ = 1

#### Multivariable Theorem

[9] x + y = y + x (Commutativ law)

[10] x * y = y * x ( Kommutativ law)

[11] x+ (y+z) = (x+y) +z = x+y+z (associativ law) [12] 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)

[14] x + xy = x (Distributive Law) [14] x + xy = x (Law Absorption)

[15] x + x’y = x + y (Law of implementation)

Description [14]

x + xy = x (1+y)

= x * 1 [6]

= x [2]

### DeMorgan’s Theorem

De Morgan Theorem is very useful for simplifying logical equations. The two theories are:

[16] (x+y)’ = x’ * y’

[17] (x*y)’ = x’ + y’

Theorem [16] 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 [17] 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.

Example.

X = [(A’+C) * (B+D’)]’

= (A’+C)’ + (B+D’)’ [17]

= (A”*C’) + (B ‘+D”) [16]

= AC’ + B’D

From the formulas [16] and [17], it can be expanded with 3 inputs and we get the following formula:

[18] (x+y+z)’ = x’ * y’ * z’

[19] (xyz)’ = x’ + y’ + z’

#### Implications of the DeMorgan Theorem

Formula [16] : (x+y)’ = x’ * y’

*Figure 1 Implications of OR to AND*

Formula [17] : (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.

### Conclusion

Logic gates can be used in electronic circuits