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.

Related Calculators | |

Boolean Algebra Calculator | Algebra Calculator |

Algebra Division | Algebra Factoring |

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.

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.

Switching In Series

**Table 1:**

**Switches in parallel**

Table 1:

Table 2: Truth Table for p or q

Table 3:**Output in terms of bits**

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

**Table 1:**

**Additive Identity**

A + 0 = A

A + 1 = 1

A + A = A

**Multiplicative Identity**

A * 0 = 0

A * 1 = A

A * A = A Below are some examples based on Boolean algebra:### Solved Examples

**Question 1: **

** Solution: **
_{}### Boolean Algebra Table

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

** Solution: **
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(x_{1}, x_{2}, x_{3}) = (x_{1} + x_{2}') . x_{3}

**Question 2: **Construct an input/output table for the Boolean function, f(x_{1}, x_{2}_{}) = (x_{1} . x'_{2})

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.

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 2:** Truth Table for P and Q

1 . 1 = 1, 1 . 0 = 0, 0 . 1 = 0, 0 . 0 = 0**Table 3:**

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.

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 = 0Table 1:

Table 2: Truth Table for p or q

Table 3:

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

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.

- 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'

A + 0 = A

A + 1 = 1

A + A = A

A * 0 = 0

A * 1 = A

A * A = A Below are some examples based on Boolean algebra:

Construct an input/output table for the Boolean function.

f(x_{1}, x_{2}, x_{3}) = (x_{1} . x_{2}') + x_{3}

The input/output table for (x_{1} . x_{2}') + x_{3} is given as follows:

Construct the required function as follows:

**Step 1:** Identify all rows of the table where the output is 1. Note that 1^{s}^{t}, 2^{n}^{d}, 3^{r}^{d} and the last row has output 1.

**Step 2:** Form the combination (x_{1}, x_{2}, x_{3}) for the rows identified in step 1.Put x_{i} if x_{i} = 1^{n}^{d} row, the expression is x_{1}.x_{2}.x_{3}'_{1}'.x_{2}'.x_{3}'

**Step 3:** Applying OR to all the combinations obtained in step 3, we have the expressionx_{1} x_{2} x_{3} + x_{1} x_{2} x_{3}' + x_{1} x_{2}' x_{3} + x_{1}' x_{2}' x_{3}'

**Step 4:** The Boolean function can be written as:f(x_{1}, x_{2}, x_{3}) = x_{1} x_{2} x_{3} + x_{1} x_{2} x_{3}' + x_{1} x_{2}' x_{3} + x_{1}' x_{2}' x_{3}'

x_{i} if x_{i} = 0

1^{s}^{t} row, the expression is x_{1}.x_{2}.x_{3}

3^{r}^{d} row, the expression is x_{1}, x_{2}'.x_{3}

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

More topics in Boolean Algebra | |

Propositional Calculus | Logical Disjunction |

Contrapositive | Boolean Algebra Examples |

Related Topics | |

Math Help Online | Online Math Tutor |