Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching...
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Transcript of Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching...
![Page 1: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/1.jpg)
Boolean Algebra
Discussion D2.2
![Page 2: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/2.jpg)
Boolean Algebra andLogic Equations
• George Boole - 1854
• Switching Algebra Theorems
• Venn Diagrams
![Page 3: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/3.jpg)
George BooleEnglish logician and mathematician
Publishes Investigation of theLaws of Thought in 1854
![Page 4: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/4.jpg)
One-variable Theorems
OR Version AND Version
X + 0 = X
X + 1 = 1
X * 1 = X
X * 0 = 0
Note: Principle of Duality You can change + to * and 0 to 1 and vice versa
![Page 5: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/5.jpg)
One-variable Theorems
OR Version AND Version
X + X' = 1
X + X = X
X * X' = 0
X * X = X
Note: Principle of Duality You can change + to * and 0 to 1 and vice versa
![Page 6: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/6.jpg)
Two-variable Theorems
• Commutative Laws
• Unity
• Absorption-1
• Absorption-2
![Page 7: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/7.jpg)
Commutative Laws
X + Y = Y + X
X*Y = Y*X
![Page 8: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/8.jpg)
Venn Diagrams
X
!X
![Page 9: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/9.jpg)
Venn Diagrams
X Y
X*Y
![Page 10: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/10.jpg)
Venn Diagrams
X + Y
X Y
![Page 11: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/11.jpg)
Venn Diagrams
X' * Y
X Y
![Page 12: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/12.jpg)
UnityX' * Y
X Y
X * Y
(X * Y) + (X' * Y) = Y
Dual: (X + Y)*(X' + Y) = Y
![Page 13: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/13.jpg)
Absorption-1
X Y
X & Y
Y + (X * Y) = Y
Dual: Y * (X + Y) = Y
![Page 14: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/14.jpg)
Absorption-2X' * Y
X Y
X + (X' * Y) = X + Y
Dual: X * (X' + Y) = X * Y
![Page 15: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/15.jpg)
Three-variable Theorems
• Associative Laws
• Distributive Laws
![Page 16: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/16.jpg)
Associative Laws
X + (Y + Z) = (X + Y) + Z
Dual:
X * (Y * Z) = (X * Y) * Z
![Page 17: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/17.jpg)
Associative Law
0 0 0 0 0 0 00 0 1 1 1 0 10 1 0 1 1 1 10 1 1 1 1 1 11 0 0 0 1 1 11 0 1 1 1 1 11 1 0 1 1 1 11 1 1 1 1 1 1
X Y Z Y + Z X + (Y + Z) X + Y (X + Y) + Z
X + (Y + Z) = (X + Y) + Z
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Distributive Laws
X * (Y + Z) = (X * Y) + (X * Z)
Dual:
X + (Y * Z) = (X + Y) * (X + Z)
![Page 19: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/19.jpg)
X Y
Z
X + (Y * Z) = (X + Y) * (X + Z)
Distributive Law - a
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Distributive Law - b
X * (Y + Z) = (X * Y) + (X * Z)
X Y
Z
![Page 21: Boolean Algebra Discussion D2.2. Boolean Algebra and Logic Equations George Boole - 1854 Switching Algebra Theorems Venn Diagrams.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649d6c5503460f94a4bbbe/html5/thumbnails/21.jpg)
Question
The following is a Boolean identity: (true or false) Y + (X * Y') = X + Y
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Absorption-2X * Y'
Y X
Y + (X * Y') = X + Y