Logic gates and Boolean.pdf
- 1. Logic gates
- 2. Basic Gates • The basic gates are AND, OR & NOT gates. • AND gate • An AND gate is a digital circuit that has two or more inputs and produces an output, which is the logical AND of all those inputs. It is optional to represent the Logical AND with the symbol ‘.’ • The following table shows the truth table of 2-input AND gate. A B Y = A.B 0 0 0 0 1 0 1 0 0 1 1 1
- 3. AND Gate A, B are the inputs and Y is the output of two input AND gate. If both inputs are ‘1’, then only the output, Y is ‘1’. For remaining combinations of inputs, the output, Y is ‘0’. The following figure shows the symbol of an AND gate, which is having two inputs A, B and one output, Y. This AND gate produces an output YY, which is the logical AND of two inputs A, B. Similarly, if there are ‘n’ inputs, then the AND gate produces an output, which is the logical AND of all those inputs. That means, the output of AND gate will be ‘1’, when all the inputs are ‘1’
- 4. OR gate • An OR gate is a digital circuit that has two or more inputs and produces an output, which is the logical OR of all those inputs. This logical OR is represented with the symbol ‘+’. A B Y = A + B 0 0 0 0 1 1 1 0 1 1 1 1
- 5. • A, B are the inputs and Y is the output of two input OR gate. If both inputs are ‘0’, then only the output, Y is ‘0’. For remaining combinations of inputs, the output, Y is ‘1’. • This OR gate produces an output YY, which is the logical OR of two inputs A, B. Similarly, if there are ‘n’ inputs, then the OR gate produces an output, which is the logical OR of all those inputs. That means, the output of an OR gate will be ‘1’, when at least one of those inputs is ‘1’.
- 6. NOT gate • A NOT gate is a digital circuit that has single input and single output. The output of NOT gate is the logical inversion of input. Hence, the NOT gate is also called as inverter. A Y = A’ 0 1 1 0
- 7. Universal gates • NAND & NOR gates are called as universal gates. Because we can implement any Boolean function, which is in sum of products form by using NAND gates alone. Similarly, we can implement any Boolean function, which is in product of sums form by using NOR gates alone • NAND gate • NAND gate is a digital circuit that has two or more inputs and produces an output, which is the inversion of logical AND of all those inputs. • The following table shows the truth table of 2-input NAND gate
- 8. A B Y = A.BA.B’ 0 0 1 0 1 1 1 0 1 1 1 0 Here A, B are the inputs and Y is the output of two input NAND gate. When both inputs are ‘1’, the output, Y is ‘0’. If at least one of the input is zero, then the output, Y is ‘1’. This is just opposite to that of two input AND gate operation. The following image shows the symbol of NAND gate, which is having two inputs A, B and one output, Y.
- 9. NOR gate • NOR gate is a digital circuit that has two or more inputs and produces an output, which is the inversion of logical OR of all those inputs. A B Y = A+BA+B’ 0 0 1 0 1 0 1 0 0 1 1 0
- 10. Here A, B are the inputs and Y is the output. If both inputs are ‘0’, then the output, Y is ‘1’. If at least one of the input is ‘1’, then the output, Y is ‘0’. This is just opposite to that of two input OR gate operation. The following figure shows the symbol of NOR gate, which is having two inputs A, B and one output, Y.
- 11. Special Gates • Ex-OR & Ex-NOR gates are called as special gates. Because, these two gates are special cases of OR & NOR gates. • Ex-OR gate • The full form of Ex-OR gate is Exclusive-OR gate. Its function is same as that of OR gate except for some cases, when the inputs having even number of ones. • The following table shows the truth table of 2-input Ex-OR gate.
- 12. A B Y = A⊕B 0 0 0 0 1 1 1 0 1 1 1 0 Here A, B are the inputs and Y is the output of two input Ex-OR gate. The truth table of Ex-OR gate is same as that of OR gate for first three rows. The only modification is in the fourth row. That means, the output YY is zero instead of one, when both the inputs are one, since the inputs having even number of ones. Therefore, the output of Ex-OR gate is ‘1’, when only one of the two inputs is ‘1’. And it is zero, when both inputs are same.
- 13. EX-NOR gate Truth Table 2- input Ex- NOR Gate B A Q 0 0 1 0 1 0 1 0 0 1 1 1 Boolean Expression Q = A ⊕ B
- 14. Logic gates using NAND gates
- 15. Logic gates using NOR gate
- 16. • 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)
- 17. • 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.
- 18. Each position represents a specific power of the base (10). For example, the decimal number 1234 consists of the digit 4 in the unit's position, 3 in the tens position, 2 in the hundreds position, and 1 in the thousands position. Its value can be written as