1. Digital Logic Systems Combinational Circuits

2. Basic Gates & Truth Tables

3. Basic Gates AND GateOR GateNOT Gate

4. More Gates NAND GateNOR GateBUF Gate

5. More Gates XNOR GateXOR Gate

6. n-Input Gates 3-Input XOR Gate 5-Input NOR Gate5-Input AND Gate 4-Input OR Gate

7. Definitions AND It gives a logical output true only if all the inputs are true OR It gives a logical output true if any of the inputs is true XOR It gives a logical output true only if an odd- number of inputs is true NOT It gives a logical output true if the input is false and vice versa

8. Truth Table A truth table is a tabular procedure to express the relationship of the outputs to the inputs of a Logical System

9. Truth Tables for Gates abf AND 000 010 100 111 abf OR 000 011 101 111 af NOT 01 10 AND OperationOR Operation NOT Operation AND GateOR GateNOT Gate

10. Truth Tables for Gates abf NAND 001 011 101 110 abf NOR 001 010 100 110 af BUF 00 11 NAND OperationNOR Operation BUF Operation NAND GateNOR GateBUF Gate

11. Truth Tables for Gates abf XOR 000 011 101 110 abf XNOR 001 010 100 111 XOR OperationXNOR Operation XNOR GateXOR Gate

12. A Bubble Implies a Logical Inversion Bubbles can be replaced by NOT Gates to get logically equivalent circuits Bubbles

22. Generate tables for all combinations of bubbles and a XOR gate

23. Gate Equivalence = = =

24. = = ?

25. = =

26. Switching Expressions

27. Basic Switching Expressions AND f = a. b OR f = a + b NOT f = a’ f = ā

28. Is there an expression for XOR operation?

31. Switching Expressions

33. f 1 = a. b’ f 2 = (a + b)’

34. Switching Expressions

36. f = ?

37. Switching Expressions f = m + n n = a’. b m = a. b’

38. Switching Expressions f = (a. b’) + (a’. b) This is the equivalent circuit and equivalent expression for a XOR operation

39. From Digital Design, 5th Edition by M. Morris Mano and Michael Ciletti

40. Switching Expressions

42. f 1 = a. b f 2 = a ^ b f 2 = (a. b’) + (a’. b)

43. Switching Expressions

44. xyzp = x ^ yg = x. ym = p. zs = p ^ zc = m + g 000 001 010 011 100 101 110 111

45. xyzp = x ^ yg = x. ym = p. zs = p ^ zc = m + g 00000 00100 01010 01110 10010 10110 11001 11101

46. xyzp = x ^ yg = x. ym = p. zs = p ^ zc = m + g 000000 001000 010100 011101 100100 101101 110010 111010

47. xyzp = x ^ yg = x. ym = p. zs = p ^ zc = m + g 00000000 00100010 01010010 01110101 10010010 10110101 11001001 11101011

48. xyzsc 00000 00110 01010 01101 10010 10101 11001 11111

49. s = s c = m + g

50. s = s c = m + g m = p. z g = g s = p ^ z

51. s = s c = m + g m = p. z g = g p = x ^ y g = x. y s = p ^ z

52. s = s c = m + g p = x ^ y g = x. y m = (x ^ y). z g = g s = (x ^ y) ^ z

53. c = ((x ^ y). z) + (x. y) p = x ^ y g = x. y m = (x ^ y). z g = g s = (x ^ y) ^ z

54. c = ((x ^ y). z) + (x. y)

55. s = ((x. y’) + (x’. y)) ^ z c = (((x. y’) + (x’. y)). z) + (x. y)

56. s = (((x. y’) + (x’. y))’. z) + (((x. y’) + (x’. y)). z’) c = (((x. y’) + (x’. y)). z) + (x. y)

57. Procedure To obtain the output functions from a logic diagram, proceed as follows: 1.Label with arbitrary symbols all gate outputs that are a function of the input variables. Obtain the Boolean Functions for each gate. 2.Label with other arbitrary symbols those gates that are a function of input variables and/or preciously labeled gates. Find the Boolean functions of these gates. 3.Repeat the process in step 2 until all the outputs of the circuit are obtained. 4.By repeated substitution of previously defined functions, obtain the output Boolean functions in terms of input variables only.