Lesson 3: Boolean Algebra and Combinational Logic

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Lesson 3 Boolean Algebra and Combinational Logic

• David L. Koehler

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Combinational Logic

• Two or more gates connected output to input.
• Defined by
• Truth table
• Boolean equation
• Logic diagram
• Present state of the outputs determined only by
  the present state of the inputs.
• A given set of input states always produces the
  same output state.
• The output state is not dependent on the order in
  which the input states are changed.

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Boolean order of operations

• AND precedes OR
• Operations within symbols of grouping are
  performed first.
• Parentheses and complement bar are symbols of
  grouping.

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Inequalities often mistaken for equalities

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DeMorgans Theorems
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The Bubble Convention

• Bubbled outputs should drive bubbled inputs.
• unBubbled outputs should drive unBubbled inputs.
• Use DeMorgan Equivalents

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Double negative

• Inversion bars of equal length over the same
  expression cancel each other.

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Sketching the logic diagram
Start inside the symbols of grouping. Do the
ANDs and then the ORs
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Sketching the logic diagram
After the ANDs, do the ORs
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Sketching the logic diagram
Outside the group, do the ANDs
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Sketching the logic diagram
Do the remaining ORs
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Making a Truth Table
Make a table with a column for each variable and
a column for each operator. Be sure to count the
NOT operator. (6 columns)
The table will have 2n1 rows where n is the
number of variables. (10 Rows)
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Making a Truth Table

• Label the leftmost columns with the names of the
  Boolean variables

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Making a Truth Table
Label one column for each complemented variable.
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Making a Truth Table
Label additional columns from left to right for
each Boolean operations observing the rules
governing the order of operations.
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Making a Truth Table
Treating the Boolean variables as bits, fill in
the sequence of binary numbers from zero to
maximum value.
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Making a Truth Table
Fill in the values for any complemented variable.
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Making a Truth Table
Substitute, evaluate, and fill in the values for
all Boolean operations.
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Deriving a Boolean Equation
Write the output of each gate having only the
Boolean variables for inputs. (Gates G1and G4.)
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Deriving a Boolean Equation
Working from input to output, fill in the
intermediate and the ultimate Boolean expressions.
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The Sum of Products (SOP)

• Any Boolean equation can be rearranged into the
  SOP form.
• A logical sum is the OR () operation.
• A logical product is the AND () operation
• An SOP can be derived from a truth table.

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SOP derived from a Truth Table

• Every HIGH output points to a MINTERM.
• Every MINTERM (AND expression) contains all
  variables in either true or complement form.
• If variable value is 0, the variable is written
  with a complement bar.
• All such MINTERMS are ORed together to form the
  SOP equation.

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SOP derived from a Truth Table
Find the rows that contain 1 in the output
column. Then write an AND expression (called a
MINTERM) for each row.
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SOP Derived from a Truth Table

• Note that the complement bar is on the
  individual variable and does not stretch across
  the AND operators.
• Note the use of the complement bars. The
  variables that are at the zero level are
  complemented.
• OR the AND expressions together.

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Simplifying the SOP equation

• Use this Boolean fact to group pairs of Minterms.
• For instance,
• Becomes

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The Product of Sums (POS)

• Any Boolean equation can be rearranged into the
  POS form.
• A logical sum is the OR () operation.
• A logical product is the AND () operation
• An POS can be derived from a truth table.

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POS derived from a Truth Table
Find the rows that contain 0 in the output
column. Then write an OR expression (called a
MAXTERM) for each row.
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POS derived from a Truth Table

• Note that the complement bar is on the
  individual variable and does not stretch across
  the OR operators.
• Note the use of the complement bars. The
  variables that are at the 1 level are
  complemented.
• AND the OR expressions together.

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Simplifying the POS

• Use this Boolean fact to group pairs
• For instance,
• Becomes

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The Karnaugh Map (K-map)

• Graphically reduce an SOP or POS.
• Just another form of Truth Table.
• Analysis can be performed from a Boolean equation
  or from Truth Table.

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Two Variable K-map
Given the SOP equation

• The 2-variable K-map looks like this

INPUTS (note the sequence)
OUTPUTS
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Two Variable K-map (SOP)
Identify groups of 1,2, and 4. Diagonals are not
allowed.
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Two Variable K-map (SOP)
Within the group of adjacent cells, retain the
variables that are the same and drop the ones
that are different.
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Two Variable K-map (POS)
Plot the MAXTERMS with 0s.
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Two Variable K-map (POS)
Look for groups of adjacent 0s.
Within the group, keep the variables the have the
same logic level. Drop the other variable.
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Three variable K-map
Count in binary Gray code.
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Three variable K-map
Groups of Four
single
Pairs
Identify all singles, and all adjacent groups of
2,4, and 8. Only vertical and horizontal
adjacencies are counted. Include as many 1s as
you can get into the fewest groups.
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Three variable K-map
Also a Pair
Single
Also a Group of Four
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Three variable K-map
This is a lesser included group.(not used)
Not a lesser include group.
The trick is to identify AS FEW groups as
possible while including all of the 1s. A larger
group trumps all lesser included groups.
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Three variable K-map (SOP)
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Three variable K-map (SOP)
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Four variable K-map
Count in binary Gray code
Count in binary Gray code.
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Four variable K-map
Illustrating additional pair forms available on
the four variable map.
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Four variable K-map
Illustrating three groups of four.
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Pick the largest groups.
The trick is to identify AS FEW groups as
possible while including all of the 1s. A larger
group trumps all lesser included groups.
This map will yield maximum simplification.
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Dont cares (adiaphorons)
Represented by X. Use them for maximum
simplification.
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Glossary

• Adjacent cell
• Associative property (any grouping order)
• Bubble to bubble convention
• Bus form
• Cell
• Combinational logic
• Cummutative property (any order)
• Distributive property of multiplication over
  addition (multiplying through)
• Dont care state
• Karnaugh map
• Levels of Gating
• Logic diagram

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Glossary

• Logic gate network
• Simplification
• Maxterm
• Minterm
• Octet
• Order of precedence
• Pair
• Product term
• Product of Sums (POS)
• Quad
• Sum of terms
• Sum of Products (SOP)
• Synthesis of a logic circuit