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Boolean Algebra

Boolean Algebra provides a basic logic for operations on binary numbers 0, 1. Since computers are based on binary system, this branch of Mathematics is found to be useful for the internal working of various computers. The application of Boolean algebra to electronic devices such as computers, lies in the restriction of the variable to two possible condition 'On and Off' or 'True or False' or numerically '1 or 0'. The electric circuits carry out the Boolean logic.

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What is Boolean Algebra?

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Boolean algebra is a logical calculus of truth values. It deals with two-values (true / false or 1 and 0) variables. In practice, electronic engineers use the symbol '1' to refer the values of the signals produced by an electronic switch as 'On' or 'True'. And, use the symbol '0' to refer the values of the signals produced by an electronic switch as 'Off' or 'False'. The symbols 0 and 1 are called bits.

Boolean Algebra Logic

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Boolean algebra is closed under AND, OR and NOT operations. A boolean operators is an operator that define relationship between a word and a group of words. But, this operators are used in logical operation with switching circuits.

Boolean Algebra Logic Gates

  • Logical OR operation
  • Logical AND operation
  • Logical NOT operation

We associate two logical operations 'AND' and 'OR' operations with switching circuits in 'series' and 'parallel' respectively.


Logical AND Operation

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Boolean Algebra Logic Gate

Let us refer to a circuit consisting of two switches p and q connected in series with a lamp and battery as shown in figure.

The lamp will glow, only if switch p and switch q are closed. If we replace the word 'closed' by T and 'open' by F, the switch will glow only if p = T and q = T. In binary language, we say the switch will glow if p = 1 and q = 1.

Table 1, Table 2 and Table 3 describes all possible states of the switches for the series connection.

Switching In Series

Table 1:

Boolean Algebra Logic Gates

Table 2: Truth Table for P and Q

Boolean Algebra Rules

The 'AND' operation can be defined on the set of bits {0, 1} as follows

1 . 1 = 1, 1 . 0 = 0, 0 . 1 = 0, 0 . 0 = 0

Table 3:

Boolean Algebra Laws

It is clear that 'AND' operation stipulated that with two input variables, the output is true only when both the inputs are true.

Logical OR Operation

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Let us refer to a circuit consisting of two switches p and q connected in parallel with a lamp and battery as shown in figure.

Boolean Algebra Law

In this case, the lamp will glow if and only if at least one of the switches is closed. In binary language, we say the switch will glow if at least one of the values of p and q is 1.

Table 1, Table 2 and Table 3 describe all possibles states of the switches for the 'OR' operation.

The 'OR' operation can be defined as the set of bits {0, 1} as follows:

1 + 1 = 1, 1 + 0 = 1, 0 + 1 = 1, 0 + 0 = 0

Switches in parallel

Table 1:

Boolean Algebra Rule


Table 2: Truth Table for p or q


Boolean Algebra Identities


Table 3:
Output in terms of bits

Boolean Algebra Identity

The OR operation stipulates that with two binary input variables, the output is true if either or both the inputs are true.

Logical NOT Operation

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The logical NOT operation is right associative and although it would produce the same result using either left or right associative property because it is a unary operator having only a single operand.

This operation has one input and one output. Table 1 represents truth table for NOT operation.

Table 1:

Boolean Algebra Logic

The 'NOT' operation is a unary operator, whereas 'AND' and 'OR' operators are binary operators.

Boolean Algebra Rules

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Boolean rules or laws is used to both reduce and simplify a complex boolean expression in an attempt to reduce the number of logic gates required. The algebra is two valued (0, 1) and has three operations (+, *, '). Boolean expression for the variable A and B is given as below:
  • A $\times$ 1 = A
  • A $\times$ 0 = 0
  • A $\times$ A = A
  • A $\times$ A' =0
  • A + 0 = A
  • A + 1 = 1
  • A + A = A
  • A + A' =1
  • A $\times$ 1 = 1
  • A $\times$ A' = 0
  • A + B = B + A ( A and B parallel to each other)
  • A $\times$ B = B $\times$ A ( A and B joined in series to each other)
  • De Morgen's Theorem states that: (A + B)' = A' $\times$ B' and (A $\times$ B)' = A' + B'

Boolean Algebra Identities

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An identity is a statement true for all possible values of its variable. The first Boolean identity is that the sum of anything and zero is the same as the original. Below you can see some basic identities of the Boolean algebra for the variable A.

Additive Identity
A + 0 = A
A + 1 = 1
A + A = A

Multiplicative Identity
A * 0 = 0
A * 1 = A
A * A = A

Boolean Algebra Examples

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Below are some examples based on Boolean algebra:

Solved Examples

Question 1:

Construct an input/output table for the Boolean function.

f(x1, x2, x3) = (x1 . x2') + x3


Solution:

The input/output table for (x1 . x2') + x3 is given as follows:

Boolean Algebra Table


Boolean Algebra Examples

Question 2: Write the Boolean expression and the Boolean function given by the input/output table as given below:

Boolean Algebra Table


Solution:
Construct the required function as follows:

Step 1: Identify all rows of the table where the output is 1. Note that 1st, 2nd, 3rd and the last row has output 1.

Step 2: Form the combination (x1, x2, x3) for the rows identified in step 1.

Put xi if xi = 1

xi if xi = 0

For the

1st row, the expression is x1.x2.x3

2nd row, the expression is x1.x2.x3'

3rd row, the expression is x1, x2'.x3

Last row, the expression is x1'.x2'.x3'

Step 3: Applying OR to all the combinations obtained in step 3, we have the expression

x1 x2 x3 + x1 x2 x3' + x1 x2' x3 + x1' x2' x3'

Step 4: The Boolean function can be written as:

f(x1, x2, x3) = x1 x2 x3 + x1 x2 x3' + x1 x2' x3 + x1' x2' x3'

Since the following is represented by a Boolean expression, it is a Boolean function.



Boolean Algebra Practice Problems

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Given below are some of the practice problems on boolean algebra.

Practice Problems

Question 1: Construct an input/output table for the Boolean function, f(x1, x2, x3) = (x1 + x2') . x3
Question 2: Construct an input/output table for the Boolean function, f(x1, x2) = (x1 . x'2)
More topics in Boolean Algebra
Propositional Calculus Logical Disjunction
Contrapositive Boolean Algebra Examples
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