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Creator QR Code 2d barcode in Software X Y AND AND gate X 0 0 1 1 Y 0 1 0 1 Truth table Z 0 0 0 1 Z

X Y AND AND gate X 0 0 1 1 Y 0 1 0 1 Truth table Z 0 0 0 1 Z
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Figure 1312 Logical multiplication and the AND gate
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Logic gate realization of the statement the output Z shall be logic 1 only when the condition (X = 1 AND Y = 1) OR (W = 1) occurs, and shall be logic 0 otherwise X 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 W 0 1 0 1 0 1 0 1 Z 0 1 0 1 0 1 1 1
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Truth table X Y AND W OR Z
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Solution using logic gates
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Figure 1313 Example of logic function implementation with logic gates
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Digital Logic Circuits
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NOT NOT gate X 1 0 X 0 1
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X 0 0 0 0 1 1 1 1
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X 1 1 1 1 0 0 0 0
Y 0 0 1 1 0 0 1 1
W 0 1 0 1 0 1 0 1
Z 0 1 1 1 0 1 0 1
(Required logic function)
Truth table for NOT gate
Figure 1314 Complements and the NOT gate
Truth table X NOT Y AND W OR Z
Solution using logic gates
Figure 1315 Solution of a logic problem using logic gates Table 1311 Rules of Boolean algebra 1 0 + X = X 2 1 + X = 1 3 X + X = X 4 X + X = 1 5 0 X = 0 6 1 X = X 7 X X = X 8 X X = 0
X 0 0 1 1 Y 0 1 0 1 (X + Y ) X(X + Y ) 0 1 1 1 0 0 1 1
Figure 1316 Proof of rule 16 by perfect induction
9 X = X 10 X + Y = Y + X 11 X Y = Y X 12 X + (Y + Z) = (X + Y ) + Z 13 X (Y Z) = (X Y ) Z 14 X (Y + Z) = X Y + X Z 15 X + X Z = X 16 X (X + Y ) = X 17 (X + Y ) (X + Z) = X + Y Z 18 X + X Y = X + Y 19 X Y + Y Z + X Z = X Y + X Z
Commutative law Associative law Distributive law Absorption law
These are stated here in the form of logic functions: (X + Y ) = X Y (X Y ) = X + Y These two laws state a very important property of logic functions: (133) (134)
Part II
Electronics
Any logic function can be implemented using only OR and NOT gates, or using only AND and NOT gates
De Morgan s laws can easily be visualized in terms of logic gates, as shown in Figure 1317 The associated truth tables are proof of these theorems
(X + Y )
Truth table X Y Z = (X + Y ) = X Y
NOT AND Z
X Y OR NOT Z
Y NOT
0 0 1 1
0 1 0 1
1 0 0 0
(XY)
X +Y
Truth table X Y Z = (X + Y ) = X + Y
NOT OR Z
Y NOT
0 0 1 1
0 1 0 1
1 1 1 0
Figure 1317 De Morgan s laws
The importance of De Morgan s laws is in the statement of the duality that exists between AND and OR operations: any function can be realized by just one of the two basic operations, plus the complement operation This gives rise to two families of logic functions: sums of products and product of sums, as shown in Figure 1318 Any logical expression can be reduced to either one of these two forms Although the two forms are equivalent, it may well be true that one of the two has a simpler implementation (fewer gates) Example 133 illustrates this point
X Y W Z
AND OR AND Sum of products expression (XY) + (WZ) (XY) + (WZ)
A B C D
OR AND OR Product of sums expression (A + B)(C + D) (A + B)(C + D)
Figure 1318 Sum-of-products and product-of-sums logic functions
13
Digital Logic Circuits
EXAMPLE 133 Simpli cation of Logical Expression
Problem
Using the rules of Table 1311, simplify the following function using the rules of Boolean algebra f (A, B, C, D) = A B D + A B D + B C D + A C D
Solution
Find: Simpli ed expression for logical function of four variables Analysis:
f =A B D+A B D+B C D+A C D =A D B +B +B C D+A C D =A D+B C D+A C D = A+A C D+B C D = A+C D+B C D =A D+C D+B C D = A D + C D (1 + B) =A D+C D = A+C D Rule 14 Rule 4 Rule 14 Rule 18 Rule 14 Rule 14 Rules 2 and 6
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