Venn Diagram – the visual aid in verifying theorems and properties

• Provides a graphical illustration of operations and relations in the algebra of sets.– The elements of a set are represented by the area

enclosed by a contour.– Given a universe N of integers from 1 to 10;

• Even numbers E = { 2, 4, 6, 8, 10 };• Odd numbers form E’s complement, = { 1, 3, 5, 7, 9 };

1

E

𝐸N = { E, }

Venn Diagram in Boolean algebra

Represent the universe B = {0, 1} by a square.• {1} using shaded area

Represent a Boolean variable x by a circle. • Area inside the circle

-> x = 1;• Area outside the circle

-> x = 0;

2

x

(a) Constant 1 (b) Constant 0

(c) Variable x (d)

x

𝑥

𝑥 𝑥

Venn Diagram – for two or more Boolean variables

Represent x, y by drawing two overlapping circles• AND operation x ∙ y -> shade overlapping area of

both circles.

-> also referred to as the

intersection of x and y.

• OR operation x + y -> shade total area within

both circles

-> also called the union of x

and y3

x y

z x

x y x y

(e) (f)

(g) (h)

x y × x y +

x y z + × x y ×

y

App: Verifying the equivalence of two expressions

4

x y

z

x y

z

x y

z

x y

z

x y

z

x y

z

x

x y × x y × x + z ×

x y z + ( ) × (a)

(d)

(c)

(f) x z ×

y z + (b)

(e)

Verification of distributive property x ∙ (y + z) = x ∙ y + x ∙ z

Another verification example

5

x y

z

y x

z

x y

z

x y × y z × z ×

x y

z

x y ×

x y

z

z

y

z

x

y

z

x

𝑥× 𝑦+𝑥×𝑧+𝑦×𝑧

𝑥× 𝑦+𝑥×𝑧

Figure 2.15. A function to be synthesized.

2.6 Synthesis using AND, OR, NOT gates

• Can express the required behavior using a truth table

6

Procedures for designing a logic circuit

• Create a product term for each valuation whose output function f is 1.– Product term: all variables are ANDed.

• Take a logic sum (OR) of these product terms to realize f.

7

f = x1x2 + + x2

f

(a) Canonical sum-of-products

f

(b) Minimal-cost realization

x 2

x 1

x 1

x 2

Figure 2.16. Two implementations of a function in Figure 2.15.8

f = x1x2 + + x2

= (x1x2 + x2)+( + x2)= (x1+ ) x2 + (+x2)= 1 ∙ x2 + ∙ 1= x2 +

Summary

• To implement a function,– Use a product term (AND gate) for each row of the

truth table for which the function is equal to 1.• If xi = 1 in the given row, xi is entered in the term;

• If xi = 0, is entered in the term.

– The sum of these product terms realizes the desired function

• Different networks can realize a given function– Use algebraic manipulation to derive simplified

logic expression, thus lower-cost networks.

9

Minterms and Sum-of-products (SOP)

• Minterms: a product term in which each of the n variables for a function appear once– Variables may appear in either un-complemented

or complemented form,

– Use mi to denote the minterm for the row number i.

• Sum-of-products Form: a logic expression consisting of product (AND) terms that are summed (ORed) – Canonical SOP: each term is a minterm

10

Figure 2.17 Three-variable minterms and maxterms.

11

Figure 2.18. A three-variable function.

12

Canonical SOP expression f = x3 + x1 +

x1+ x1

Manipulate f as followingf = (x1+ ) x3 +

x1(x2+ )

= 1 x3 + x1 1

= x3 + x1

A more concise form to specify the given canonical SOP expression (logical sum)

f =

Maxterms and Product-of-Sums (POS)

• Maxterms: complements of minterms– By applying the principle of duality, if we could

synthesize a function f by considering the rows for which f = 1, it should also be possible to synthesize f by considering the rows where f = 0

• Product-of-sums Form: a logic expression consisting of sum (OR) terms that are the factors of a logical product (AND) – Canonical POS: each term is maxterm

13

Figure 2.17 Three-variable minterms and maxterms.

14

An example

• The complement of a function can be represented by a sum of minterms for which f = 0.

• Complement this expression using DeMorgan’s theorem = M2

15

Figure 2.18. A three-variable function.

16

= m0 + m2 + m3 + m7

f = = = M0 M2 M3 M7

= (x1+x2+x3) (x1++x3) (x1++) (++)

f = (x1+x3) (+)

A more concise form to specify the given canonical POS expression (logical product)

f =

Figure 2.19. Two realizations of a function in Figure 2.18.

f

(a) A minimal sum-of-products realization

x1

x2

x3

• Cost of a logic circuit is – the total number of gates plus– the total number of inputs to all gates in the circuit.

17

f = x3 + x1

Cost = 13

Figure 2.19. Two realizations of a function in Figure 2.18.

f

(a) A minimal sum-of-products realization

f

(b) A minimal product-of-sums realization

x1

x2

x3

x2

x1x3

18

Cost = 13

Example 2.3

Consider the function

f(x1,x2,x3) =

1. Canonical SOP expression for the function

f = m2+m3+m4+m6+m7

=

2. Simplify the expression

f = +

= +

19

Example 2.4

Consider the function in Example 2.3, Specify it as a product of maxterms for which f = 0

f(x1,x2,x3) =

1. Canonical POS expression for the function

f = M0M1M5

= (x1+x2+x3)(x1+x2+)(+x2+)

2. Simplify the expression

f = (x1+x2)(x2+)

= x2+x1 20

Discussion (1)

Given a logic function f(x1,x2,x3),

1. What is the index of the maxterm ?

) ->

0 1 0

(010)2 = 2 (decimal number)

Therefore, = m2

2. What is the logic expression of m5?

(5)10 = (1 0 1)2

=> m5 = ( 21

Complemented entry -> 0uncomplement entry -> 1

Discussion (2)

Given a logic function f(x1,x2,x3),

1. What is the index of the maxterm ) ?

) ->

1 0 1 (101) = 5 (decimal)

Therefore, ) = M5

2. What is the logic expression of M5?

(5)10 = (1 0 1)2

=> M5 = (22

Complemented entry -> 1uncomplement entry -> 0

Venn Diagram for Boolean algebra

• Basic requirement for legal Venn diagram– Must be able to represent all minterms of a

Boolean function

23

Two variables

x1 x2

m1

m0

m3m2

Three variables

m0

m7

x1x2

x3

m2

m6

m3m5

m4

m1

Venn Diagram for Boolean algebra

• Basic requirement for legal Venn diagram– Must be able to represent all minterms of a

Boolean function

24

Two variables

x1 x2

m1

m0

m3m2

Three variables

m0

m7?

x1x2 x3

m2

m6 m3

m5?

m4 m1