boolean algebra is a mathematical tool that deals with logical operations and binary number system. it builds the foundation of digital electronics and computer science.

the laws and rules in boolean algebra are the sets of logical statements or expressions upon which all the logical expressions are built. each law of the boolean algebra can be interpreted as an operation performed by a logic circuit like a logic gate.

in this chapter, we will learn about laws and rules of boolean algebra that are used to simplify the logical functions and boolean expressions. these laws and rules are essential tools in boolean algebra that help to reduce the complexity and optimize the digital circuits and systems.

let us learn the primary laws and rules of boolean algebra in detail that are used to perform logical operations.

laws of boolean algebra

all the important laws and rules of boolean algebra are explained below −

rules of logical operations

there are three basic logical operations namely, and, or, and not. the following table highlights the rules associated with these three logical operations −

and operation	or operation	not operation
0 and 0 = 0	0 or 0 = 0	not of 0 = 1
0 and 1 = 0	0 or 1 = 1	not of 1 = 0
1 and 0 = 0	1 or 0 = 1
1 and 1 = 1	1 or 1 = 1

these rules of boolean algebra can be implemented using logic gates.

and laws

in boolean algebra, there are four and laws given below −

law 1 − a · 0 = 0 (this law is called null law).
law 2 − a · 1 = a (this law is called identity law).
law 3 − a · a = a
law 4 − a · a' = 0

or laws

there are four or laws described below −

law 1 − a + 0 = a (this law is called null law).
law 2 − a + 1 = 1 (this law is called identity law).
law 3 − a + a = a
law 4 − a + a' = 1

complementation laws

there are following five complementation laws in boolean algebra −

law 1 − 0' = 1
law 2 − 1' = 0
law 3 − if a = 0, then a' = 1
law 4 − if a = 1, then a' = 0
law 5 − (a')' = a (this is called double complementation law)

commutative laws

there are following two commutative laws in boolean algebra −

law 1 − according to this law, the operation a or b produces the same output as the operation b or a, i.e.,

a + b = b + a

hence, the order of the variables does not affect the or operation.

this law can be extended to any number of variables. for example, for three variables, it will be,

a + b + c = c + b + a = b + c + a = c + a + b

law 2 − according to this law, the output of the a and b operation is same as that of the b and a operation, i.e.,

a · b = b · a

this law states that the order in which the variables are anded does not affect the result.

we can extend this law to any number of variables. for example, for three variables, we get,

a · b · c = a · c · b = c · b · a = c · a · b

associative laws

associative laws define the ways of grouping the variables. there are two associative laws as described below.

law 1 − the expression a or b ored with c results the same as the a ored with b or c, i.e.,

(a + b) + c = a + (b + c)

this law can be extended to any number of variables. for example, for 4 variables, we get,

(a + b + c) + d = a + (b + c + d) = (a + b) + (c + d)

law 2 − the expression a and b anded with c results the same as the expression a anded with b and c, i.e.,

(a · b) · c = a · (b · c)

we can extend this law to any number of variables. for example, if we have 4 variables, then

(abc)d = a(bcd) = (ab)·(cd)

distributive laws

in boolean algebra, there are the following two distributive laws that allow for multiplying or factoring out of expressions.

law 1 − according to this law, we or several variables and then and the result with a single variable.

it gives the same result as the expression in which the single variable is anded with each of the several variables and then ored the product terms, i.e.,

a · (b + c) = ab + ac

we can extend this law to any number of variables. for example,

a(bc + de) = abc + ade

ab(cd + ef) = abcd + abef

law 2 − according to this law, if we and several variables and then the result is ored with a single variable.

it gives the same result as we or the single variable with each of the several variables and then the sum terms are anded together, i.e.,

a + bc = (a + b)(a + c)

proof − the proof of this law is explained here,

rhs = (a + b)(a + c)

= aa + ab + ac + bc

= a + ab + ac + bc

= a (1 + b + c) + bc

since,

1 + b + c = 1 + c = 1

therefore,

a · 1 + bc = a + bc = lhs

redundant literal rule (rlr)

under this rule, there are two laws in boolean algebra, which are explained here.

law 1 − according to this law, if we or a variable with the and of the complement of the variable and another variable. then, it is same as the or of the two variables, i.e.,

a + ab = a + b

proof − the proof of this law is explained here,

lhs = a + ab = (a + a)(a + b)

= 1 · (a + b) = a + b = rhs

law 2 − according to this law, if we and a variable with the or of the complement of the variable and another variable, it is equivalent to when we and the two variables, i.e.,

a(a + b) = ab

proof − this law can be proved as follows,

lhs = a(a + b) = aa + ab

= 0 + ab = ab = rhs

both these laws show that the complement of a term appearing in another term is redundant. hence, the rule is named as redundant literal rule.

idempotence laws

the term "idempotence" is a synonym for "same value". there are two idempotence laws in boolean algebra. they are,

law 1 − according to this law, anding a variable with itself is equal to the variable, i.e.,

a · a = a

law 2 − according to this law, oring a variable with itself is equal to the variable, i.e.,

a + a = a

absorption laws

there are two absorption laws in boolean algebra and they are explained below.

law 1 − according to this law, if we or a variable with the and of the that variable and another variable, then it is equal to the variable itself, i.e.,

a + a · b = a

this can be proved as follows,

lhs = a + a · b = a · (1 + b)

= a · 1 = a = rhs

law 2 − according to this law, the and of a variable with the or of that variable and another variable is equivalent to the variable itself i.e.,

a(a + b) = a

this can also be proved as follows,

lhs = a(a + b) = aa + ab

= a + ab = a(1 + b) = a · 1 = a = rhs

hence, this law proves that if a term appears in another term, then the latter term will become redundant and can be removed from the expression.

demorgan's theorem

in boolean algebra, demorgans theorem defines two laws which are explained below.

law 1 − according to this law, the complement of a sum of variables is equivalent to the product of complement of each of the variables, i.e.,

$$\mathrm{\overline{a+b} \: = \: \bar{a}\cdot\bar{b}}$$

this law can be extended to any number of variables.

law 2 − the second law of demorgans theorem states that the complement of a product of variables is equivalent to the sum of complement of each of the variables, i.e.,

$$\mathrm{\overline{ab} \: = \: \bar{a}\: + \:\bar{b}}$$

this law can also be extended to any number of variables.

conclusion

in this chapter, we explained all the important laws, rules, and theorems used in boolean algebra. these rules and laws are extensively used to simplify the logical expressions in digital electronics.

basically, all these rules provide a set of tools for simplification of complex boolean functions and make the digital circuits simpler.

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