Boolean
Algebra
Module
3
|
Reading
Assignment (4th edition):
|
Problems, 1, 2(A,D,F), 4 |
|
Boolean Algebra
- Algebras, binary logic, Boolean expressions, and truth tables
- Evaluation of Boolean expressions
- Basic gate symbols: AND, OR, NAND, NOR, INVERT, XOR, XNOR
- Logical multiplication (AND), logical addition (OR) and unary operator (complement)
Boolean Functions with Two Input Variables
- Four input combinations {00,01,10,11}
- Sixteen possible functions
Input |
F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 |
00 01 10 11 |
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
- Which
of the preceding correspond to known Boolean functions?
- Functions of more than two input variables?
- Precedence rules and the evaluation of Boolean functions
- Parentheses à NOT à AND à OR
- Examples
Combinational Networks
- Gate symbols and DeMorgan's
equivalents, and DeMorgan’s square
·
OR equivalent
found by using DeMorgan's Theorem: x + y = (x'y')' = (x')'+(y')' = x + y
·
Alternatively AND
equivalent found by using DeMorgan's Theorem: xy = (x' + y')' = (x')'(y')' = xy
- Description
and properties of combinational networks
·
Side-effect free
·
Propagation
delays
·
Design approach:
Boolean expressions, algebraic simplification, gate level design
Circuit Analysis
- Given
a circuit generate the truth table (switching function) and boolean
expression (switching expression)
- Examples
Circuit Synthesis
- Given
a switching (boolean) expression construct the gate network
- Examples
Principal identities in Boolean Algebra
|
|
|
|
|
0 + a = a |
1a = a |
Identity |
|
1 + a = 1 |
0a = 0 |
|
|
a + a = a |
aa = a |
Idempotency |
|
a + a' = 1 |
aa' = 0 |
Complement |
|
(a')' = a |
|
Involution |
|
ab + a’c + bc = ab + a’c |
(a+b)(a’+c)(b+c) = (a+b)(a’+c) |
Consensus Theorem |
Equivalent expressions and algebraic simplification
- Duality Principle
- A Boolean expression remains valid if we take
the dual of expressions on the LHS and RHS
- Replace 0s by 1s and
ANDs by ORs and vice versa
- Principal identities can be used to simplify boolean expressions
- Examples
Universal Gate Sets
- What gate types do we need to realize arbitrary boolean functions?
- AND, OR, NOT : more natural
- We really need only AND and NOT or OR and NOT
- NAND or NOR : single gate type
Standard Forms
- Minterms and maxterms: definitions in truth table
- minterm à 1's
- maxterm à 0's
- representing minterms with input variables, e.g., ABC
- representing maxterms with input variables, e.g., (A + B + C)
- Sum of minterms
- unique, decimal-notation
- For example, F(A,B,C) = m1 + m3 + m6 = ∑ m(1,3,6) = one-set(1,3,6)
- canonical sum of products (SOP)
- implementing a function as sum of minterms
- construction of sum of minterms from boolean expressions: algebraic vs. truth table
- example
- Product of maxterms
- unique, decimal-notations
- For example, F(A,B,C) = M1 + M3 + M6 = ∏ M(1,3,6) = zero-set(1,3,6)
- canonical product of sums (POS)
- implementing a function as product of maxterms
- construction of product of maxterms from boolean expressions: algebraic vs. truth table
- example
- Converting between Minterm form and Maxterm form
Sum of Products and Product of Sums
- Sum of minterms corresponds to SOP: example
- Different levels
- 1-level à both POS and SOP
- ex: A+B+C = (A·1)+(B·1)+(C·1) = (A+B+C)·(1)
- ex: A·B·C = (A+0)·(B+0)·(C+0) = (A·B·C)+0
- 2-level à either POS or SOP
- ex: (A+B)·(B+C) à POS
- ex: (A·B)+(B·C) à SOP
- SOP can always be rewritten as POS and vice versa
- 3-level à neither POS or SOP
- ex: (A+B)·(C+B·D)
- ex: A·C+A·B+A·(C+B)
- can always be rewritten into 2nd level form by Boolean reduction
- Implementation in all NAND or NOR gates
- POS example
- best when implemented with all NOR gates
- SOP example
- best when implemented with all NAND gates
Examples
Example - How Minterms are Represented by Input Variables
Example - How Maxterms are Represented by Input Variables
Example - Sum of Minterms Example
Thus, the final function, F, may be written as the OR (or sum) of these minterm functions. i.e. F=*=(A'·B'·C)+(A'·B·C')+(A·B·C')
Example - Product of Maxterms Example
Thus, the final function, F, may be written as the AND (or product) of these maxterm functions. i.e. F=*=(A+B+C)(A+B'+C')(A'+B+C')
Example - How Sum of Minterms Corresponds to SOP
distributive
property using by multiplying the third term by A + A’ = 1
complement law, B + B’ = 1
F and Freduced are equivalent functions. While F is the sum of minterms, both F and Freduced are sum of products.
Example:
Implement the following POS function using only NOR gates. Assume you have all signal and signal complements.
Solution:
Now replace all gates with NOR gates or DeMorgan equivalent NOR gates. Notice, by choosing NOR gates for implementing POS, inverters are required at the input only.
Example:
Implement the following SOP function using only NAND gates. Assume you have all signal and signal complements.
Solution:
Now replace all gates with NAND gates or DeMorgan equivalent NAND gates. Notice, by choosing NAND gates for implementing SOP, inverters are required at the input only.
