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Relationship Between Basic Operation of Boolean and Basic Logic Gate The basic construction of a logical circuit is gates Gate is an electronic circuit that emits an output signal as a result of a simple Boolean operation on its inputs Logical function is presented through the combination of gates The basic gates used in digital logic is the same as the basic Boolean algebra operations (e.g., AND, OR, NOT,…)

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The package Truth Tables and Boolean Algebra set out the basic principles of logic. A B F 0 0 0 0 1 0 1 0 0 1 1 1 A B F 0 0 0 0 1 1 1 0 1 1 1 1 A B F 0 0 0 0 1 1 1 0 1 1 1 1 A B F 0 0 0 0 1 0 1 0 0 1 1 1 F F F F Name Graphic Symbol Boolean Algebra Truth Table ABAB ABAB ABAB ABAB A F AND OR NOT NAND NOR F = A. B Or F = AB F = A + B _____ F = A + B ____ F = A. B Or F = AB _ F = A B F 0 1 1 0 the symbols, algebra signs and the truth table for the gates

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1. Identity Elements 2. Inverse Elements 1. A = A A. A = 0 0 + A = A A + A = 1 3. Idempotent Laws 4. Boundess Laws A + A = A A + 1 = 1 A. A = A A. 0 = 0 5. Distributive Laws 6. Order Exchange Laws A. (B + C) = A.B + A.C A. B = B. A A + (B. C) = (A+B). (A+C) A + B = B + A 7. Absorption Laws 8. Associative Laws A + (A. B) = AA + (B + C) = (A + B) + C A. (A + B) = AA. (B. C) = (A. B). C 9. Elimination Laws 10. De Morgan Theorem A + (A. B) = A + B (A + B) = A. B A. (A + B) = A. B (A. B) = A + B Basic Theorems of Boolean Algebra

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Exercise 1 Apply De Morgan theorem to the following equations: F = V + A + L F = A + B + C + D Verify the following expressions: S.T + V.W + R.S.T = S.T + V.W A.B + A.C + B.A = A.B + A.C

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Relationship Between Boolean Function and Logic Circuit A B Q Boolean function Q = AB + B = (NOT A AND B) OR B Logic circuit A AB B = AB + B

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Relationship Between Boolean Function and Logic Circuit Any Boolean function can be implemented in electronic form as a network of gates called logic circuit A B F A.B = AB C D C + D = AB + C + D

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G = A. (B + C + D) A B C D C + D B + C + D

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Truth Table

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A B Q A AB B = AB + B Produce a truth table from the logic circuit ABAABQ 00100 01111 10000 11001

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G = A. (B + C + D) Exercise 2 Build a truth table for the following Boolean function

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Karnaugh Map A graphical way of depicting the content of a truth table where the adjacent expressions differ by only one variable For the purposes simplification, the Karnaugh map is a convenient way of representing a Boolean function of a small number (up to four) of variables The map is an array of 2 n squares, representing all possible combination of values of n binary variables Example: 2 variables, A and B A B 0001 1011 B A B A B A B A 10 1 0

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00000001 0100 1100 1000 AB C D A B C D A B CD A B C D 4 variables, A, B, C, D 2 4 = 16 squares

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000010110100 001011111101 AB C C C 000001 010011 110111 100101 ABC C 00011110 0101 00 01 11 10 01 List combinations in the order 00, 01, 11, 10 C

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ABCF 0001 0010 0100 0111 1001 1011 1100 1110 Truth Table Karnaugh Map 11 11 0 00 11 11 0 BC A 0 1 A A How to create Karnaugh Map 1.Place 1 in the corresponding square

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11 0 00 11 11 0 AB F = AB + AB A B Karnaugh Maps to Represent Boolean Functions

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2.Group the adjacent squares: Begin grouping square with 2 n-1 for n variables e.g. 3 variables, A, B, and C 2 3-1 = 2 2 = 4 = 2 1 = 2 = 2 0 = 1 11 11 0 00 11 11 0 BC A 0 1 A A AB BC ABC F = BCABABC + +

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1 1111 0 00 11 11 0 BC A 0 1 A A 3 variables: 2 3-1 = 2 2 = 4 2 2-1 = 2 1 = 2 2 1-1 = 2 0 = 1 A BC F = A + BC

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11 1 1 111 AB 01 00 CD 11 10 11 4 variables, A, B, C, D 2 4-1 = 2 3 = 8 ( maximum ); 2 2 = 4; 2 1 = 2; 2 0 = 1 ( minimum ); CD +BDABC+F =

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The following diagram illustrates some of the possible pairs of values for which simplification is possible:

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Karnaugh Map Boolean Function Logic Circuit

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Transform the following truth table to Karnaugh Map and find the Boolean function Exercise 3

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