1. Boolean Algebra – Part 1
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2. Boolean Algebra Objectives
Understand Basic Boolean Algebra
Relate Boolean Algebra to Logic Networks
Prove Laws using Truth Tables
Understand and Use First Basic Theorems
Apply Boolean Algebra to:
Simplifying Expressions
Multiplying Out Expressions
Factoring Expressions
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3. A New Kind of Algebra
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4. Truth Tables
A truth table provides a complete enumeration of the inputs and the corresponding output for a function.
If there are n inputs,
There will be 2n rows
In the table.
Unlike with regular algebra, full enumeration is possible
(and useful) in Boolean Algebra
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5. Complement Operation Also known as invert or not.
The inverse of x is written as x’ or x.
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6. Logical AND Operation “•” denotes AND
Output is true iff all inputs are true
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7. Logical OR Operation “+” denotes OR
Output is true if any inputs are true
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8. Boolean Expressions
Boolean expressions are made up of variables and constants combined by AND, OR and NOT
Examples: A’ A•B C+D AB A(B+C) AB’+C
• A•B is the same as AB (• is omitted when obvious).
• Parentheses are used like in regular algebra for grouping.
• NOT has highest precedence, followed by AND then OR.
A term is a grouping of variables ANDed together.
A literal is each instance of a variable or constant.
This expression has 4 variables, 3 terms, and 6 literals:
a’b + bcd + a
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9. Boolean Expressions
Each Boolean expression can be specified by a truth table which lists all possible combinations of the values of all variables in the expression.
F = A’ + B C
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10. Boolean Expressions From Truth Tables
Each 1 in the output of a truth table specifies one term in the corresponding boolean expression.
The expression can be read off by inspection…
F is true when:
A is false AND B is true AND C is false OR
A is true AND B is true AND C is true
F = A’BC’ + ABC
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11. Another Example
F = ?
F = A’B’C +
A’BC’ +
AB’C’ +
ABC
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12. There may be multiple expressions for any given truth table.
Yet Another Example
F = ?
F = A’B’ + A’B + AB’ + AB = 1 (F is always true)
There may be multiple expressions for any given truth table.
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13. Converting Boolean Functions to Truth Tables
F = AB + BC BC AB BC
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14. Converting Boolean Functions to Truth Tables
F = AB + BC BC AB BC
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15. Some Basic Boolean Theorems
From these…
We can derive
these…
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16. Proof Using Truth Tables
Truth Tables can be used to prove that 2 Boolean expressions are equal.
If the two expressions have the same value for all possible combinations of input variables, they are equal.
X Y’ + Y = X + Y =
These two expressions are equal.
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17. Basic Boolean Algebra Theorems
Here are the first 5 Boolean Algebra theorems we will study and use.
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18. Basic Boolean Algebra Theorems
While these laws don’t seem very exciting, they can be very useful in simplifying Boolean expressions:
Simplify: (M N’ + M’ N) P + P’ + 1 X
+ 1 1
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19. Commutative Laws Associative Laws
X • Y = Y • X X + Y = Y + X
Associative Laws
(X • Y) • Z = X • (Y • Z) = X • Y • Z
(X + Y) + Z = X + (Y + Z ) = X + Y + Z
Just like regular algebra
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20. Just like regular algebra
Distributive Law
X ( Y + Z ) = X Y + X Z
Prove with a truth table: =
Just like regular algebra
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21. Other Distributive Law
X + Y Z = ( X + Y ) ( X + Z )
( X + Y ) ( X + Z ) = X ( X + Z) + Y ( X + Z )
= X X + X Z + Y X + Y Z
= X + X Z + X Y + Y Z
= X • 1 + X Z + X Y + Y Z
= X ( 1 + Z + Y ) + Y Z
= X • 1 + Y Z
= X + Y Z
Proof:
NOT like regular algebra!
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22. Simplification Theorems
X Y + X Y’ = X ( X + Y ) (X + Y’) = X
X + X Y = X X ( X + Y ) = X
X + Y X’ = X + Y X ( X’ + Y ) = X Y
These are useful for simplifying Boolean Expressions.
The trick is to find X and Y.
( A’ + B + CD) (B’ + A’ + CD)
( A’ + CD + B) (A’ + CD + B’)
Rearrange terms
A’ + CD
(X + Y)(X + Y’) = X
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23. Multiplied out = sum-of-products form (SOP)
Multiplying Out
All terms are products only
(no parentheses)
A B C’ + D E + F G H Yes
A B + C D + E Yes
A B + C ( D + E ) No
Multiplied out = sum-of-products form (SOP)
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24. Multiplying Out - Example
( A’ + B )( A’ + C )( C + D )
Use ( X + Y )( X + Z ) = X + Y Z
( A’ + B C )( C + D )
Multiply Out
A’ C + A’ D + B C C + B C D
Use X • X = X
A’ C + A’ D + B C + B C D
Use X + X Y = X
A’ C + A’ D + BC
Using the theorems may be simpler than brute force.
But, brute force does work…
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25. Factoring Final form is products only
All sum terms are single variables only
( A + B + C’ ) ( D + E ) Yes
( A + B )( C + D’ )E’F Yes
( A + B + C’ ) ( D + E ) + H No
( A’ + BC ) ( D + E ) No
Factored out = product-of-sums form (POS)
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26. Factoring - Example A B + C D ( A B + C )( A B + D )
Sum of Products (SOP)
A B + C D
Use X + Y Z = ( X + Y )( X + Z )
( A B + C )( A B + D )
Use X + Y Z = ( X + Y )( X + Z ) again
( A + C )( B + C )( A B + D )
And again
( A + C )( B + C )( A + D )( B + D )
Product of Sums (POS)
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27. POS vs. SOP Any expression can be written either way
Can convert from one to another using theorems
Sometimes SOP looks simpler
AB + CD = ( A + C )( B + C )( A + D )( B + D )
Sometimes POS looks simpler
(A + B)(C + D) = BD + AD + BC + AC
SOP will be most commonly used in this class but we must learn both
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28. Duality If an equality is true its dual will be true as well
To form the dual:
AND  OR
Invert constant 0’s or 1’s
Do NOT invert variables
Because these are true
These are also true
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29. For each theorem on the left, the dual is listed on the right.
Theorem Summary
For each theorem on the left, the dual is listed on the right.
X + 0 = X X • 1 = X
X + 1 = 1 X • 0 = 0
X + X = X X • X = X
X + X’ = 1 X • X’ = 0
(X’)’ = X
XY + XY’ = X (X + Y) (X + Y’) = X
X + XY = X X(X + Y) = X
X + YX’ = X + Y X(Y + X’) = X Y
XY + X’Z + YZ = XY + X’Z (X + Y)(X’ + Z)(Y + Z) = (X + Y)(X’ + Z)
HINT: when forming duals, first apply parentheses around all AND terms.
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