Digital logic gates and Boolean algebra


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Digital logic gates and Boolean algebra

  2. 2. LOGIC GATESINTRODUCTION: A logic gate is an electronic circuit/device which makes logic decisions. Most logic gates are two inputs and one outputs. At any given moment, every terminal is in one of the two binary conditions low (0) or high(1), represented by different voltage levels.
  3. 3.  The logic state of a terminal can, and generally does, change often as the circuit processes data. In most logic gates, the low state is approximately 0v, while the high state is approximately 5v. Logic gates are also called as switches. with the advent of integrate circuits, switches have been replaced by TTL circuit and CMOS circuits. symbolic logic uses values, variables and operations.
  4. 4. TYPES OF LOGIC GATES:The most common logic gates used are,Basic gates1.OR2.AND3.NOTUniversal gates1.NAND2.NOR X-OR or Exclusive-OR
  5. 5. OR GATE: The OR gate has two or more inputs and one output. Its output is true if at least one input is true.SYMBOL:
  6. 6.  The OR operation may be defined as “Y equals A OR B”. Y=A+B Where, the symbol ‘+’ indicates the OR concept. Each terminal may assume two possible values either zero or one.
  7. 7. TRUTH TABLE: A B A+B 0 0 0 0 1 1 1 0 1 1 1 1
  8. 8. AND GATE: The AND gate is also a basic kind of digital circuit. It has also two or more inputs and one output.SYMBOL:
  9. 9.  The AND operation for the output is defined as, “y equals A AND B”. Y=A.B Where ‘.’ symbol indicates AND operation. The output of the AND gate is one only when both inputs are one.
  10. 10. TRUTH TABLE: A B A+B 0 0 0 0 1 0 1 0 0 1 1 1
  11. 11. NOT GATE or Inverter Gate: A NOT gate is a basic gate that has one input and one output.SYMBOL:
  12. 12.  The NOT circuit serves to invert the polarity of any input pulse apply to it. If A is the input then output “Y equals to NOT A or Ā. Y= Ā Where, the bar symbol over A represents NOT or compliment operation
  13. 13. TRUTH TABLE: A Ā 0 1 1 0
  14. 14. NAND GATE: The NAND gate is known as an universal gate because it can be used to realize all the three basic functions of OR, AND & NOT gates. It is also called as NOT-AND gate.SYMBOL:
  15. 15.  The Boolean expression for the NAND operation is given by, Y=A.B
  16. 16. TRUTH TABLE: A B AB 0 0 1 0 1 1 1 0 1 1 1 0
  17. 17. NOR GATE: The NOR gate is also a universal gate and it is a combination of a NOT and OR gates.SYMBOL:
  18. 18.  The Boolean expression for NOR gate is given by, Y=A+B
  19. 19. TRUTH TABLE: A B A+B 0 0 1 0 1 0 1 0 0 1 1 0
  20. 20. Exclusive OR or X-OR GATE: The X-OR gate is a logic gate having two inputs with and single output.SYMBOL:
  21. 21.  The Boolean expression for the X-OR gate is given by, Y=A+B+ Where + indicates the exclusive OR + operation and in terms of expression it can be expanded as Y=AB+AB
  22. 22. TRUTH TABLE: A B AB+AB 0 0 0 0 1 1 1 0 1 1 1 0
  23. 23. ADVANTAGES OF LOGIC GATES: It is generally very easy to reliably distinguish between logic 1 or logic 0. The simplest flip-flop is the RS which is made up of two gates. K-map is also designed by using logic gates. That simplification helps when you start to connect gates to implement the functions. These gates are also used in TTL and CMOS circuitary.
  24. 24. BOOLEAN ALGEBRA Boolean Algebra derives its name from the mathematician George Boole in 1854 in his book “An investigation of the laws of taught”. Instead of usual algebra of numbers Boolean algebra is the algebra of truth values 0 or 1. In order to fully understand this the relation between the AND gate, OR gate & NOT gate operations should be appreciated.
  25. 25. POSTULATES OF BOOLEAN ALGEBRA: The Boolean algebra has its own set of fundamental laws which differ from the ordinary algebra. They are,OR laws: A+0=A A+1=1 A+A=A A+Ā=1
  26. 26. AND laws: A.0=0 A.A=A A.1=A A.Ā=0NOT laws: 0=1 1=0 If A=0 then Ā=1 If A=1 then Ā=0 Ā=A
  27. 27. Commutative law: A+B=B+A A.B=B.AAssociative laws: A+(B+C)=(A+B)+C A.(B.C)=(A.B).C (A+B)+(C+D)=A+B+C+D
  28. 28. Distributive laws: A.(B+C)=(A.B)+(A.C) (A+B).C=A.C+B.C A+ĀB=A+B A+B.C=(A+B).(A+C)Absorptive laws: A+A.B=A A.(A+B)=A A.(Ā+B)=ABDemorgan’s laws: A+B=A.B A.B=A+B
  29. 29. EXAMPLE: (AB+C)(AB+D)=AB+CD AB.AB+AB.D+C.AB+C.D AB+ABD+ABC+CD {A.A=A} AB(1+D)+ABC+CD {1+A=1} AB+ABC+CD AB(1+C)+CD AB+CD
  30. 30. Advantages: If we use Boolean algebra for your logical problem you can save more gates and operations. so your design will be cheaper, more comprehensible, more serviceable . It allows logical steps quickly and repeatedly.Disadvantages: Can only arrive at direct results not implied once.