week 2.1 boolean algebra - pami.uwaterloo.capami.uwaterloo.ca/~basir/ece124/week2-1.pdf ·...

WEEK 2.1 BOOLEAN ALGEBRA

1

Boolean Algebra

Boolean algebra was introduced in 1854 by George Boole and in 1938 was shown by C. E. Shannon to be useful for manipulating Boolean logic functions.

The postulates and theorems of Boolean algebra are useful to simplify expressions, to prove equivalence of expressions, etc.

ECE 124 Digital Circuits and Systems

Axioms/Postulates of Boolean Algebra (1) Boolean algebra is an algebraic structure defined by a set of elements, B, together with two binary

operators + and * that satisfy the following postulates:

1. Postulate 1:

a) Closure with respect to +.

b) Closure with respect to *. 2. Postulate 2:

a) An identity element with respect to +, designated by 0.

b) An identity element with respect to *, designated by 1. 3. Postulate 3:

a) Commutative with respect to +.

b) Commutative with respect to *. 4. Postulate 4:

a) Distributive * over +.

b) Distributive + over *. 5. Postulate 5:

For each element x in B, there is an element : x in B (the complement) such that (a) x+ !x = 1 and (b) x * !x = 0.

6. Postulate 6:

There exists at least two elements x,y in B, such that x y.


Axioms/Postulates of Boolean Algebra (2) Using the definitions of AND/OR and NOT functions, we can show all the

postulates are satisfied.


Note: Postulates are facts that can be taken as true; they do not require proof.

Theorems and Properties of Boolean Algebra Theorems help us out in manipulating Boolean expressions.

They must be proven from the postulates and/or other theorems known to be proven correct.


Note the duality in relationships if we interchange + with * and 0 with 1.

DeMorgan:

6

31

2

U?A

32

12

U?A

05

12

U?A

05

31

2

U?A

32

12

U?A

05

12

U?A

05

1

23

U?A

00𝑥 + 𝑦 = 𝑥′. 𝑦′ ′

(𝑥 . 𝑦 ) = 𝑥′ + 𝑦′ ′

(𝑥′. 𝑦′) = 𝑥 + 𝑦′

(𝑥′ + 𝑦 ) = 𝑥 . 𝑦′ ′

Synthesis

• While Boolean algebra is good and truth tables tell us what a function

does, how do we actually make or implement a function?

– The implementation of a circuit is known as synthesis and we can draw

circuits implementing functions using the logic gate symbols.

• Consider two functions and how we can implement them with gates.


𝑓1 = 𝑥′. 𝑦′. 𝑧 + 𝑥. 𝑦′. 𝑧′ + 𝑥. 𝑦′. 𝑧 + 𝑥. 𝑦. 𝑧′ + 𝑥. 𝑦. 𝑧

How can we use three input and gate and inverters

to produce a desirable function output?

REMEMBER: AND produces o/p of 1 only if all i/p’s are 1.

𝑓2 = 𝑥′. 𝑦′. 𝑧 + 𝑥′. 𝑦. 𝑧 + 𝑥. 𝑦′. 𝑧′ + 𝑥. 𝑦′. 𝑧

Synthesis Example for f1 and f2


• Circuit for f1:

• Circuit for f2:

Circuit cost for f1 and f2


• For a circuit, we will define a COST. We will ignore NOT gates at the inputs of a circuit;

all other gates count as 1 and every gate input will count as 1.

• Cost of f1 is 1 AND + 1 OR + 4 GATE INPUTS = 6.

• Cost of f2 is 3 AND + 1 OR + 11 GATE INPUTS = 15.

Synthesis Example for f2 (Again)


• This new circuit also implements f2 and its cost is 2 AND + 1 OR + 6 GATE

INPUTS = 9.

• The cost of the circuit is a measure of how expensive it will be to implement (in this

case, we compute cost that measures the area of the circuit).

• So, why is Boolean algebra important? Well, if we can get a simpler expression for a

function, we can get a cheaper and simpler circuit (this is good!!!)

Terminology

• Basic terminology…

• We have seen that Boolean algebra, logic, etc. works with binary variables.

• The variables have values in {0,1}; E.g., x in {0,1} and x is a binary variable.

• It is sometimes convenient to refer to the un-complemented (i.e., x) and complemented (i.e., !x) versions of a variable x.

• These two terms are called literals. – The un-complemented version x is called the positive literal of variable x. – The complemented version !x is called the negative literal of variable x.

• So, for each binary variable, there are two associated literals.


Minterms and Maxterms

• A truth table for an n-input function will have 2n rows.

• Minterms: – For each row of the truth table, create an AND of the literals according to the following

rule: If a variable has value 1 in the row, include its +ve literal. If a variable x has value 0 in the row, include its –ve literal.

– The resulting AND of each literal is called a MINTERM.

• Maxterms – For each row of the truth table, create an OR of the literals according to the fllowing rule:

If a variable x has value 0 in the row, include its +ve literal. If a variable x has value 1 in the row, include its –ve literal.

– The resulting OR of each literal is called a MAXTERM.

• So minterms and maxterms are created “opposite” of each other.


Illustration of Minterms and Maxterms.


• Minterms are denoted by a lower case “m” while Maxterms are denoted by an upper

case “M”.

• When a particular input pattern appears, its associated minterm will be 1 while all

other minterms will evaluate to 0.

• When a particular input pattern appears, its associated maxterm will be 0 while all

other maxterms will evaluate to 1.

• Minterms and maxterms are “duals” of each other; !mi = Mi and !Mi = mi

Canonical Sum-of-Products (SOP)


• Given a truth table, we can ALWAYS write a logic expression for the

function by taking the OR of the minterms for which the function is a 1.

– This representation of a function is a “sum of minterms” and is called a

canonical sum-of-products (SOP) representation of the function.

• Examples:

• Shortcut notation:

Sum-of-Products Implementations

• If implemented with gates, a SOP will always have the following form. – A plane of NOT gates (inverters to generate all literals), followed by…

– A plane of AND gates (to implement the minterms), followed by…

– A single OR gate (to take the “sum”).


Canonical Product-of-Sums (POS)


• Given a truth table, we can ALWAYS write a logic expression for the function

by taking the AND of the maxterms for which the function is a 0.

– This representation of a function is a “product of maxterms” and is called a canonical

product-of-sums (POS) representation of the function.

• Examples:

• Shortcut notation:

Product-Of-Sums Gate Implementations

• If implemented with gates, a POS will always have the form: – A plane of NOT gates (inverters to generate all literals), followed by…

– A plane of OR gates (to implement the maxterms), followed by…

– A single AND gate (to take the “product”).


General Comments

• There are always two canonical representations for a function, the SOP or the POS.

• Sometimes, one implementation is simpler than the other implementation (in terms of its cost).

• SOP and POS implementations are often referred to as 2-level logic implementations.

– This is because we assume NOT gates at the input are free, so we see that there are two levels of gates (AND-OR for SOP and OR-AND for POS) required to implement the function.


Conversion between SOP and POS

• It is always possible to convert between a POS and SOP representation for a functin.

• Consider f1= (1,4,7) which can also be expressed as f1 = (0,2,3,5,6).

f1 = (1,4,7)

= m1+m4+m7

= !(!f1) // double inversion is okay

= !(m0 + m2 + m3 + m5 + m6) // !f1 is those minterms not in f1

= (!m0)(!m2)(!m3)(!m5)(!m6)

= (M0)(M2)(M3)(M5)(M6)

= (0,2,3,5,6).

• Note: Quickly, we can from minterms (maxterms) to maxterms (minterms) by changing () to () and list those indices of terms missing from the original list.


Standard Sum-Of-Products (1)

• A function described using a canonical SOP (minterms) is by no means minimal. It might require more gates/literals than required.

• Let us call any AND of literals a product term.

• We can then express logic functions in Standard Sum-Of-Products form where, instead of minterms, the AND terms are simply product terms.

• We can start with a canonical SOP and use Boolean algebra to simply the expression into something simpler.


Standard Sum-Of-Products (2)


• Let’s consider one of our previous functions in Canonical SOP.

• If we used Boolean algebra to simplify, we would find that f2 can also

be written as a Standard SOP using a sum of product terms:

Standard Product-Of-Sums (1)

• A function described using a canonical POS (maxterms) is by no means minimal. It might require more gates/literals than required.

• Let us call any OR of literals a sum term.

• We can then express logic functions in Standard Product-Of-Sums form where, instead of maxterms, the OR terms are simply sum terms.

• We can start with a canonical POS and use Boolean algebra to simply the expression into something simpler.


Example of Standard Product-Of-Sums Forms

• Let’s consider one of our previous functions in Canonical POS.


• If we used Boolean algebra to simplify, we would find that f1 can also

be written as a Standard POS using a product-of-sum terms:

Other Logic Gates

• Although we can always implement any function we want using AND/OR/NOT, there are other types of logic gates that prove useful.


NAND and NOR gates (2-inputs)

NAND gate performs a “NOT-AND” operation.


• NAND/NOR gates can be extended to multiple inputs, but the NAND/NOR gates are

not associative (explained later). We should always think of NAND as “NOT-AND”

and NOR as “NOT-OR”.

• NOR gate performs a “NOT-OR” operation.

NAND and NOR gates (n-inputs)

• Think of multiple input NAND/NOR gates in terms of the operations they perform; i.e., NOT-AND (for a NAND) and NOT-OR (for a NOR).

• Example: 3-input versions:


XOR and NXOR gates (2-inputs)

• XOR gate (with 2-inputs performs a “difference operation”):


• XOR/NXOR gates are incredibly useful for arithmetic operations like

addition/subtraction/multiplication. These gates can also be extended to multiple

inputs, but we need to be clear on their definitions with multiple inputs.

• NXOR gate (with 2-inputs performs a “equivalence operation”):

XOR gates with multiple inputs.

• A XOR gate with > 2 inputs performs the “odd operation”; the output is a 1 whenever an odd number of inputs are 1.



• XOR gates are associative (explained later).

NXOR gates with multiple inputs.

• A NXOR gate with > 2 inputs performs the “odd function”; the output is a 1 whenver an odd number of inputs are 1.



• NXOR gates are non-associative (explained later).

Buffer (1-input)

• Does nothing logically; Used in implementation to “boost” a signal’s strength.


Associative and Non-Associative Gates

AND/OR gates are associative gates. This means that we can collapse

many smaller AND (OR) gates into a single AND (OR) gate with

multiple inputs.

Example:


• XOR gates are also associative. Not all types of logic gates are associative.

Non-Associative Gates

NAND/NOR and NXOR gates are not

associative.


Example 1: Proof of a Theorem

• Prove Absorption Theorem 6(a)

• Can prove using postulates, or using truth tables.

• Solution:


Example 2: Expression Simplification

• Simply the following logic expression:

• Solution …

• Note: Could have also asked “Show that” (i.e., LHS = RHS):


week 2.1 boolean algebra - pami.uwaterloo.capami.uwaterloo.ca/~basir/ece124/week2-1.pdf ·...

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