STUDY RESOURCES: A thorough overview of the different Boolean algebra laws

Boolean algebra is the mathematics used to analyze various circuits and gates in digital electronics. The laws of Boolean algebra can be used to simplify and reduce a handful number of complex Boolean expressions in an honest attempt to reduce the required number of logic gates.

The variables used in Boolean algebra have only two possible values:

i) Logic 0,

ii) Logic 1.

Meaning, if an expression is made of three variables (say, X, Y, and Z) with the form of X + Y = Z, each of the variables can possibly have only 2 values (either 0 or 1).

Truth tables of the different Boolean Algebra laws

BOOLEAN EXPRESSION	DESCRIPTION
X + 1 = 1	X is in parallel with CLOSED
X + 0 = X	X is in parallel with OPEN
X . 1 = X	X is in series with CLOSED
X . 0 = 0	X is in series with OPEN
X + X = X	X is in parallel with X
X . X = X	X is in series with X
_ X + X = 1	X is in parallel with NOT X
_ X + X = 0	X is in series with NOT X
X + Y = Y + X	X is in parallel to Y = Y is in parallel to X
X . Y = Y . X	X is in series with Y = Y is in series with X
__ __      _ _ X + Y = X . Y	INVERT. OR is replaced with AND (AKA De Morgan’s law**)
_  __     _ _ X. Y = X + Y	INVERT. AND is replaced by OR (AKA De Morgan’s law**)

**De Morgan’s laws:

De Morgan’s laws states that”

• The complement of the union of 2 sets is exactly same as that of the intersection of the 2 complements.

• The complement of the intersection of 2 sets is exactly same as that of the union of their complements.

(The symbolic representation is provided in the table above.)

Do note that the Boolean algebraic laws mentioned above are mainly represented by a single or at most two variable. But this is not fixed in any way whatsoever. An infinite number of variables can be taken as inputs to the expressions provided above.

A brief description of the laws provided above (with X as user input) is provided below for your reference.

Descriptions of the different Boolean algebraic laws

• Annulment law

Any term AND-ed with 0 equals 0.

Any term OR-ed with 1 equals 1.

For e.g.,

X . 0 = 0

X + 1 = 1

• Identity Law

Any term OR-ed with 0 or AND-ed with 1 will always equal the same term.

For e.g.,

X + 0 = X

X . 1 = X

• Idempotent Law

Any input that is AND-ed or OR-ed with the same gives a result that is equal to the input itself.

For e.g.,

X + X = X

X . X = X

• Complement Law

Any term AND-ed with its immediate complement equals 0.

Any term OR-ed with its immediate complement equals 1.

For e.g.,
_
X . X = 0
_
X + X = 1

• Commutative law

The order of arrangement of two individual terms is unimportant for altering the results.

For e.g.

X . Y = Y. X

X + Y = Y + X

• Double negation law

Any term that’s inverted twice is equal to its original term.

For e.g.,

• De Morgan’s law

It’s already explained above. Please refer to the same. We would just provide the expression again for your reference:

A few other Boolean algebraic laws

• Distributive Law:

X ( Y + Z) = X . Y + X . Z

X + (Y . Z) = (X + Y) . (X + Z)

• Absorptive Law:

X + (X . Y) = X

X (X + Y) = X

• Associative law:

X + (Y + Z) = (X + Y) + Z = X + Y + Z

X (Y . Z) = (X. Y) Z = X . Y . Z

So that’s it then. We hope you found the laws exceptionally handy for your school or college assignments. With that, we’ll bring this to a close for now. Adieu!