Boolean Functions

A Boolean function refers to a function having n number of entries or variables, so it has 2n number of possible combinations of the given variables. Such functions would only assume 0 or 1 in their output. An example of a Boolean function is, f(p,q,r) = p X q + r. We are implementing these functions with the logic gates.

In this article, we will take a look at Boolean Functions according to the GATE Syllabus for CSE (Computer Science Engineering). Read ahead to learn more.

Table of Contents

• What are Boolean Functions?
• Truth Table Formation
• Methods to Simplify the Boolean Function
  • Karnaugh-map or K-map
    • Sum of Products (SOP) Form
    • Product of Sums (POS) Form
  • NAND Gates Realization
• Practice Problems on Boolean Algebra
• Video on Logic Gates & Minimisation of Boolean Functions
• FAQs

What are Boolean Functions?

In Boolean algebra, we deal with various logic operations and binary variables. Here, we make use of an algebraic expression known as the Boolean Expression to describe the Boolean Function. The Boolean Expression contains the logic operation symbols, binary variables, and the constants 1 and 0. Let us now consider the example given below:

F (W, X, Y, Z) = W + XȲ + WZY Equation No. 1

The left side of this equation here represents the output B. So we can state equation no. 1

B = W + XȲ + WZY

Truth Table Formation

We use a truth table to represent a table that has all the combinations of inputs along with their corresponding results. Conversion of the switching equation to the truth table is also possible. For instance, let us consider the switching equation given as follows:

F (W, X, Y) = W + XY

The output here would be high (1) when W = 1 or XY = 1 or when both of these are 1. For this equation, we get the truth table by Table (a). The total number of rows present in the truth table would be 2n, where n refers to the total number of input variables (for the given equation, n=3). Hence, there are a total of 23 = 8 possible input combinations.

Methods to Simplify the Boolean Function

Here are the methods that we used in order to simplify the Boolean function:

• NAND gate method
• K-map or Karnaugh-map

Karnaugh-map or K-map

The De-Morgan’s theorems and the Boolean theorems are useful in manipulating the given logic expression. One can use gates to realize the logical expression. The total number of logic gates that we require for the realization of any logical expression must be reduced to its minimum possible value by the K-map method. We can perform this method in two different ways and have discussed these methods below. Read ahead to know more.

Sum of Products (SOP) Form

If we have three expressions, then these expressions are known to be in their SOP form when they are in the form of a sum of the three terms WX, WY, XY, where all the individual terms are a product of the two variables (like W.X or W.Y, etc.). Therefore, these are known as Sum of Products or SOP. Remember that the products and the sum in SOP form are NOT at all the actual multiplications or additions. In fact, these are actually the AND and OR functions. In the SOP form, 1 represents an unbar and 0 represents a bar. The SOP form is basically represented by .

Here is an example of SOP for your understanding:

Product of Sums (POS) Form

If we have three expressions, then these expressions are known to be in their POS form when they are in the form of a product of three terms (A+B), (X+Y), or (W+Y), where every term is in the form of the sum of the two variables. Therefore, these expressions are called Products of Sums form or the POS form. In the POS form, 1 represents a bar and 0 represents an unbar. The POS form is basically represented by .

Here is an example of POS for your understanding:

NAND Gates Realization

We can use the NAND Gates to simplify the given Boolean functions. We have shown this in the example below. Check it out.

Practice Problems on Boolean Algebra

1. We use the _________ to implement any Boolean function.

a) Arithmetic logics

b) Logical notations

c) Expressions

d) Logic gates

Answer – (d) Logic gates

2. We perform the inversion of a single bit input to output using the _________ gate.

a) NAND

b) AND

c) NOR

d) NOT

Answer – (d) NOT

3. Which of these refers to a Boolean variable?

a) String

b) Literal

c) Identifier

d) Keyword

Answer – (b) Literal

4. The minimization of the function F(X,Y,Z) = X*Y*(Y+Z) is _________:

a) Y+Z

b) XZ

c) XY

d) Y`

Answer – (c) XY

Video on Logic Gates & Minimisation of Boolean Functions

FAQs

Q1

What is a Boolean function? Explain with examples.

A Boolean function refers to a function having n number of entries or variables, so it has 2n number of possible combinations of the given variables. Such functions would only assume 0 or 1 in their output. An example of a Boolean function is, f(p,q,r) = p X q + r. We are implementing these functions with the logic gates.

Q2

What are the different Boolean functions?

Various Boolean functions have three or more inputs. The most common ones out of these are OR, AND, XOR, NOR, XNOR, and NAND. The N-input AND gate would produce a TRUE output only when all of the N inputs happen to be TRUE. On the other hand, the N-input OR gate would produce a TRUE output only when at least one of the inputs happens to be TRUE.

Q3

How do you define the Boolean function?

One can define the Boolean function F=ab’ c+p in terms of four of the binary variables a, b, c, and p. Such a function would be equal to 1 when a=1, b=0, or c=1. Also, apart from an algebraic expression, we can also describe the Boolean function in terms of the truth table.

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