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Logic gates ppt

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This document discusses Boolean algebra and logic gates. It begins by defining basic logic gates like AND, OR, NOT, NAND, NOR, and XOR. It then provides truth tables and circuit diagrams for each gate. The document also covers Boolean algebra concepts like Boolean constants, variables, functions, and theorems. Additional topics discussed include Boolean laws, converting between logic circuits and equations, adding binary numbers, and applications to computer memory and processing.

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Boolean Algebra Logic Gates 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 1
2 Basic logic gates Not And Or Nand Nor Xor x x x y xy x y xyz z x+yx y x y x+y+z z x y xy x+yx y xyx y 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
The AND gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 3
The OR gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 4
The NOT gate (or inverter) 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 5
A logic buffer gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 6
The NAND gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 7
The NOR gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 8
The Exclusive OR gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 9
The Exclusive NOR gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 10
Boolean Algebra Boolean Constants these are 0 (false) and 1 (true) Boolean Variables variables that can only take the vales 0 or 1 Boolean Functions each of the logic functions (such as AND, OR and NOT) are represented by symbols as described above Boolean Theorems a set of identities and laws see text for details 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 11
Boolean laws ABBA BAAB ++ ))(( )( CABABCA BCABCBA +++ ++ CBACBA CABBCA ++++ )()( )()( ABAA AABA + + )( BABA BABA +緒 件+ ABBAA BABAA + ++ )( 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 12
OR Gate Current flows if either switch is closed Logic notation A + B = C A B C 0 0 0 0 1 1 1 0 1 1 1 1 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 13
AND Gate In order for current to flow, both switches must be closed Logic notation AB = C (Sometimes AB = C) A B C 0 0 0 0 1 0 1 0 0 1 1 1 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 14
Properties of AND and OR Commutation o A + B = B + A o A B = B A Same as Same as 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 15
Commutation Circuit A + B B + A A B B A 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 16
Properties of AND and OR Associative Property A + (B + C) = (A + B) + C A (B C) = (A B) C = 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 17
Distributive Property (A + B) (A + C) A B C Q 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 18
Binary Addition A B S C(arry) 0 0 0 0 1 0 1 0 0 1 1 0 1 1 0 1 Notice that the carry results are the same as AND C = A B 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 19
Inversion (NOT) A Q 0 1 1 0 Logic: AQ 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 20
Circuit for XOR Accumulating our results: Binary addition is the result of XOR plus AND BABABA +獣 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 21
22 Converting between circuits and equations Find the output of the following circuit Answer: (x+y)y Or (xy)y x y x+y y (x+y)y __ x y y 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
23 x y Converting between circuits and equations Find the output of the following circuit Answer: xy Or (xy) xy x y x y x y _ ____ 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
24 Converting between circuits and equations Write the circuits for the following Boolean algebraic expressions a) x+y x y x y x y __ x x+y 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
25 x y x y x y Converting between circuits and equations Write the circuits for the following Boolean algebraic expressions b) (x+y)x _______ x y x+y x+y (x+y)x 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
26 Writing xor using and/or/not p q (p q) 測(p q) x y (x + y)(xy) x y xy 1 1 0 1 0 1 0 1 1 0 0 0 x y x+y xy xy (x+y)(xy) ____ 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
27 Converting decimal numbers to binary 53 = 32 + 16 + 4 + 1 = 25 + 24 + 22 + 20 = 1*25 + 1*24 + 0*23 + 1*22 + 0*21 + 1*20 = 110101 in binary = 00110101 as a full byte in binary 211= 128 + 64 + 16 + 2 + 1 = 27 + 26 + 24 + 21 + 20 = 1*27 + 1*26 + 0*25 + 1*24 + 0*23 + 0*22 + 1*21 + 1*20 = 11010011 in binary 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
28 Converting binary numbers to decimal What is 10011010 in decimal? 10011010 = 1*27 + 0*26 + 0*25 + 1*24 + 1*23 + 0*22 + 1*21 + 0*20 = 27 + 24 + 23 + 21 = 128 + 16 + 8 + 2 = 154 What is 00101001 in decimal? 00101001 = 0*27 + 0*26 + 1*25 + 0*24 + 1*23 + 0*22 + 0*21 + 1*20 = 25 + 23 + 20 = 32 + 8 + 1 = 41 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
29 A note on binary numbers In this slide set we are only dealing with non- negative numbers The book (section 1.5) talks about twos- complement binary numbers Positive (and zero) twos-complement binary numbers is what was presented here We wont be getting into negative twos- complmeent numbers 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
30 How to add binary numbers Consider adding two 1-bit binary numbers x and y 0+0 = 0 0+1 = 1 1+0 = 1 1+1 = 10 Carry is x AND y Sum is x XOR y The circuit to compute this is called a half-adder x y Carry Sum 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
31 The half-adder Sum = x XOR y Carry = x AND y x y Sum Carry x y Sum Carry x y Carry Sum 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
32 Using half adders We can then use a half-adder to compute the sum of two Boolean numbers 1 1 0 0 + 1 1 1 0 010? 001 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
33 How to fix this We need to create an adder that can take a carry bit as an additional input Inputs: x, y, carry in Outputs: sum, carry out This is called a full adder Will add x and y with a half-adder Will add the sum of that to the carry in What about the carry out? Its 1 if either (or both): x+y = 10 x+y = 01 and carry in = 1 x y c carry sum 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 0 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
34 HAX Y S C HAX Y S C x y c c s HAX Y S C HAX Y S C x y c The full adder The HA boxes are half-adders x y c s1 c1 carry sum 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 s1 c1 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
35 The full adder The full circuitry of the full adder x y s c c 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
36 Adding bigger binary numbers Just chain full adders together HAX Y S C FAC Y X S C FAC Y X S C FAC Y X S C x1 y1 x2 y2 x3 y3 x0 y0 s0 s1 s2 s3 c ... 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
37 Adding bigger binary numbers A half adder has 4 logic gates A full adder has two half adders plus a OR gate Total of 9 logic gates To add n bit binary numbers, you need 1 HA and n-1 FAs To add 32 bit binary numbers, you need 1 HA and 31 FAs Total of 4+9*31 = 283 logic gates To add 64 bit binary numbers, you need 1 HA and 63 FAs Total of 4+9*63 = 571 logic gates 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
38 More about logic gates To implement a logic gate in hardware, you use a transistor Transistors are all enclosed in an IC, or integrated circuit The current Intel Pentium IV processors have 55 million transistors! 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
39 Flip-flops Consider the following circuit: What does it do? 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
40 Memory A flip-flop holds a single bit of memory The bit flip-flops between the two NAND gates In reality, flip-flops are a bit more complicated Have 5 (or so) logic gates (transistors) per flip-flop Consider a 1 Gb memory chip 1 Gb = 8,589,934,592 bits of memory Thats about 43 million transistors! In reality, those transistors are split into 9 ICs of about 5 million transistors each 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
41 Hexadecimal A numerical range from 0-15 Where A is 10, B is 11, and F is 15 Often written with a 0x prefix So 0x10 is 10 hex, or 16 0x100 is 100 hex, or 256 Binary numbers easily translate: 4/30/2018 Pavithran Puthiyapurayil , Maldives National University

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Logic gates ppt

  • 1. Boolean Algebra Logic Gates 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 1
  • 2. 2 Basic logic gates Not And Or Nand Nor Xor x x x y xy x y xyz z x+yx y x y x+y+z z x y xy x+yx y xyx y 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 3. The AND gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 3
  • 4. The OR gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 4
  • 5. The NOT gate (or inverter) 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 5
  • 6. A logic buffer gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 6
  • 7. The NAND gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 7
  • 8. The NOR gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 8
  • 9. The Exclusive OR gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 9
  • 10. The Exclusive NOR gate 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 10
  • 11. Boolean Algebra Boolean Constants these are 0 (false) and 1 (true) Boolean Variables variables that can only take the vales 0 or 1 Boolean Functions each of the logic functions (such as AND, OR and NOT) are represented by symbols as described above Boolean Theorems a set of identities and laws see text for details 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 11
  • 13. OR Gate Current flows if either switch is closed Logic notation A + B = C A B C 0 0 0 0 1 1 1 0 1 1 1 1 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 13
  • 14. AND Gate In order for current to flow, both switches must be closed Logic notation AB = C (Sometimes AB = C) A B C 0 0 0 0 1 0 1 0 0 1 1 1 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 14
  • 15. Properties of AND and OR Commutation o A + B = B + A o A B = B A Same as Same as 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 15
  • 16. Commutation Circuit A + B B + A A B B A 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 16
  • 17. Properties of AND and OR Associative Property A + (B + C) = (A + B) + C A (B C) = (A B) C = 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 17
  • 18. Distributive Property (A + B) (A + C) A B C Q 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 18
  • 19. Binary Addition A B S C(arry) 0 0 0 0 1 0 1 0 0 1 1 0 1 1 0 1 Notice that the carry results are the same as AND C = A B 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 19
  • 20. Inversion (NOT) A Q 0 1 1 0 Logic: AQ 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 20
  • 21. Circuit for XOR Accumulating our results: Binary addition is the result of XOR plus AND BABABA +獣 4/30/2018 Pavithran Puthiyapurayil , Maldives National University 21
  • 22. 22 Converting between circuits and equations Find the output of the following circuit Answer: (x+y)y Or (xy)y x y x+y y (x+y)y __ x y y 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 23. 23 x y Converting between circuits and equations Find the output of the following circuit Answer: xy Or (xy) xy x y x y x y _ ____ 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 24. 24 Converting between circuits and equations Write the circuits for the following Boolean algebraic expressions a) x+y x y x y x y __ x x+y 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 25. 25 x y x y x y Converting between circuits and equations Write the circuits for the following Boolean algebraic expressions b) (x+y)x _______ x y x+y x+y (x+y)x 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 26. 26 Writing xor using and/or/not p q (p q) 測(p q) x y (x + y)(xy) x y xy 1 1 0 1 0 1 0 1 1 0 0 0 x y x+y xy xy (x+y)(xy) ____ 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 27. 27 Converting decimal numbers to binary 53 = 32 + 16 + 4 + 1 = 25 + 24 + 22 + 20 = 1*25 + 1*24 + 0*23 + 1*22 + 0*21 + 1*20 = 110101 in binary = 00110101 as a full byte in binary 211= 128 + 64 + 16 + 2 + 1 = 27 + 26 + 24 + 21 + 20 = 1*27 + 1*26 + 0*25 + 1*24 + 0*23 + 0*22 + 1*21 + 1*20 = 11010011 in binary 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 28. 28 Converting binary numbers to decimal What is 10011010 in decimal? 10011010 = 1*27 + 0*26 + 0*25 + 1*24 + 1*23 + 0*22 + 1*21 + 0*20 = 27 + 24 + 23 + 21 = 128 + 16 + 8 + 2 = 154 What is 00101001 in decimal? 00101001 = 0*27 + 0*26 + 1*25 + 0*24 + 1*23 + 0*22 + 0*21 + 1*20 = 25 + 23 + 20 = 32 + 8 + 1 = 41 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 29. 29 A note on binary numbers In this slide set we are only dealing with non- negative numbers The book (section 1.5) talks about twos- complement binary numbers Positive (and zero) twos-complement binary numbers is what was presented here We wont be getting into negative twos- complmeent numbers 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 30. 30 How to add binary numbers Consider adding two 1-bit binary numbers x and y 0+0 = 0 0+1 = 1 1+0 = 1 1+1 = 10 Carry is x AND y Sum is x XOR y The circuit to compute this is called a half-adder x y Carry Sum 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 31. 31 The half-adder Sum = x XOR y Carry = x AND y x y Sum Carry x y Sum Carry x y Carry Sum 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 32. 32 Using half adders We can then use a half-adder to compute the sum of two Boolean numbers 1 1 0 0 + 1 1 1 0 010? 001 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 33. 33 How to fix this We need to create an adder that can take a carry bit as an additional input Inputs: x, y, carry in Outputs: sum, carry out This is called a full adder Will add x and y with a half-adder Will add the sum of that to the carry in What about the carry out? Its 1 if either (or both): x+y = 10 x+y = 01 and carry in = 1 x y c carry sum 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 0 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 34. 34 HAX Y S C HAX Y S C x y c c s HAX Y S C HAX Y S C x y c The full adder The HA boxes are half-adders x y c s1 c1 carry sum 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 s1 c1 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 35. 35 The full adder The full circuitry of the full adder x y s c c 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 36. 36 Adding bigger binary numbers Just chain full adders together HAX Y S C FAC Y X S C FAC Y X S C FAC Y X S C x1 y1 x2 y2 x3 y3 x0 y0 s0 s1 s2 s3 c ... 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 37. 37 Adding bigger binary numbers A half adder has 4 logic gates A full adder has two half adders plus a OR gate Total of 9 logic gates To add n bit binary numbers, you need 1 HA and n-1 FAs To add 32 bit binary numbers, you need 1 HA and 31 FAs Total of 4+9*31 = 283 logic gates To add 64 bit binary numbers, you need 1 HA and 63 FAs Total of 4+9*63 = 571 logic gates 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 38. 38 More about logic gates To implement a logic gate in hardware, you use a transistor Transistors are all enclosed in an IC, or integrated circuit The current Intel Pentium IV processors have 55 million transistors! 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 39. 39 Flip-flops Consider the following circuit: What does it do? 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 40. 40 Memory A flip-flop holds a single bit of memory The bit flip-flops between the two NAND gates In reality, flip-flops are a bit more complicated Have 5 (or so) logic gates (transistors) per flip-flop Consider a 1 Gb memory chip 1 Gb = 8,589,934,592 bits of memory Thats about 43 million transistors! In reality, those transistors are split into 9 ICs of about 5 million transistors each 4/30/2018 Pavithran Puthiyapurayil , Maldives National University
  • 41. 41 Hexadecimal A numerical range from 0-15 Where A is 10, B is 11, and F is 15 Often written with a 0x prefix So 0x10 is 10 hex, or 16 0x100 is 100 hex, or 256 Binary numbers easily translate: 4/30/2018 Pavithran Puthiyapurayil , Maldives National University