Logic Gates & Boolean operations | IBPS PO Prelims & Mains Preparation - Bank Exams PDF Download

Logic Gates

The logic gates are the main structural part of a digital system.

• Logic Gates are a block of hardware that produces signals of binary 1 or 0 when input logic requirements are satisfied.
• Each gate has a distinct graphic symbol, and its operation can be described by means of algebraic expressions.
• The seven basic logic gates includes: AND, OR, XOR, NOT, NAND, NOR, and XNOR.
• The relationship between the input-output binary variables for each gate can be represented in tabular form by a truth table.
• Each gate has one or two binary input variables designated by A and B and one binary output variable designated by x.

The AND gate is an electronic circuit which gives a high output only if all its inputs are high. The AND operation is represented by a dot (.) sign.

The OR gate is an electronic circuit which gives a high output if one or more of its inputs are high. The operation performed by an OR gate is represented by a plus (+) sign.

The NOT gate is an electronic circuit which produces an inverted version of the input at its output. It is also known as an Inverter.

The NOT-AND (NAND) gate which is equal to an AND gate followed by a NOT gate. The NAND gate gives a high output if any of the inputs are low. The NAND gate is represented by a AND gate with a small circle on the output. The small circle represents inversion.

The NOT-OR (NOR) gate which is equal to an OR gate followed by a NOT gate. The NOR gate gives a low output if any of the inputs are high. The NOR gate is represented by an OR gate with a small circle on the output. The small circle represents inversion

The 'Exclusive-OR' gate is a circuit which will give a high output if one of its inputs is high but not both of them. The XOR operation is represented by an encircled plus sign.

The 'Exclusive-NOR' gate is a circuit that does the inverse operation to the XOR gate. It will give a low output if one of its inputs is high but not both of them. The small circle represents inversion.

Boolean algebra 

Boolean algebra can be considered as an algebra that deals with binary variables and logic operations. Boolean algebraic variables are designated by letters such as A, B, x, and y. The basic operations performed are AND, OR, and complement. 

The Boolean algebraic functions are mostly expressed with binary variables, logic operation symbols, parentheses, and equal sign. For a given value of variables, the Boolean function can be either 1 or 0.

Basics of Combinatorial Components

Combinational circuit is a circuit in which we combine the different gates in the circuit, for example encoder, decoder, multiplexer and demultiplexer.

• Half Adder 
  • Half adder is a combinational logic circuit with two inputs and two outputs. The half adder circuit is designed to add two single bit binary numbers A and B. It is the basic building block for addition of two single bit numbers. This circuit has two outputs carry and sum.
• Full Adder
  • Full adder is developed to overcome the drawback of Half Adder circuit. It can add two one-bit numbers A and B, and carry c. The full adder is a three input and two output combinational circuit. 
• Multiplexers 
  • Multiplexer is a special type of combinational circuit. There are n-data inputs, one output and m select inputs with 2m = n. It is a digital circuit which selects one of the n data inputs and routes it to the output. 
• Demultiplexers 
  • A demultiplexer performs the reverse operation of a multiplexer i.e. it receives one input and distributes it over several outputs. It has only one input, n outputs, m select input. 
• Decoder 
  • A decoder is a combinational circuit. It has n input and to a maximum m = 2n outputs. Decoder is identical to a demultiplexer without any data input. It performs operations which are exactly opposite to those of an encoder. 
• Encoder 
  • Encoder is a combinational circuit which is designed to perform the inverse operation of the decoder. An encoder has n number of input lines and m number of output lines. 
• Priority Encoder 
  • This is a special type of encoder. Priority is given to the input lines. If two or more input line are 1 at the same time, then the input line with highest priority will be considered.

Number System

When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. A computer can understand the positional number system where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.

The value of each digit in a number can be determined using −

• The digit
• The position of the digit in the number
• The base of the number system (where the base is defined as the total number of digits available in the number system)

Decimal Number System
The number system that we use in our day-to-day life is the decimal number system. Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represent units, tens, hundreds, thousands, and so on.

Binary Number System 
Characteristics of the binary number system are as follows − 

• Uses two digits, 0 and 1 
• Also called as base 2 number system 
• Each position in a binary number represents a 0 power of the base (2). Example 20 
• Last position in a binary number represents a x power of the base (2). Example 2x where x represents the last position - 1. 

Octal Number System
Characteristics of the octal number system are as follows − 

• Uses eight digits, 0,1,2,3,4,5,6,7 
• Also called as base 8 number system 
• Each position in an octal number represents a 0 power of the base (8). Example 80 
• Last position in an octal number represents a x power of the base (8). Example 8x where x represents the last position – 1 

Hexadecimal Number System: 
Characteristics of hexadecimal number system are as follows − 

• Uses 10 digits and 6 letters, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F 
• Letters represent the numbers starting from 10. A = 10. B = 11, C = 12, D = 13, E = 14, F = 15 
• Also called as base 16 number system 
• Each position in a hexadecimal number represents a 0 power of the base (16). Example, 160 
• Last position in a hexadecimal number represents a x power of the base (16). Example 16x where x represents the last position – 1.