Equal Boolean Functions - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

Download Equal Boolean Functions - Discrete Mathematics - Lecture Slides and more Slides Discrete Mathematics in PDF only on Docsity!

Discrete Mathematics

CS 2610

September 17, 2008

2

Equal Boolean Functions

Two Boolean functions F and G of degree n are

equal iff for all (x 1 ,..xn ) ∈ Bn^ , F (x1 ,..xn ) = G (x1 ,..xn )

Example: F(x,y,z) = x(y+z), G(x,y,z) = xy + xz, and F=G (recall the “truth” table we put on the board yesterday) Also, note the distributive property: x(y+z) = xy + xz via the distributive law

4

Boolean Functions

More equivalent Boolean expressions: (x + y)z = xyz + xyz + xyz (x + y)z = xz + yz distributive = x1z + 1yz identity = x(y + y)z + (x + x)yz unit = xyz + xyz + xyz + xyz distributive = xyz + xyz + xyz idempotent

We’ve expanded the initial expression into its sum of products form.

5

Boolean Functions

More equivalent Boolean expressions: xy + z = ?? = xy1 + 11z identity = xy(z + z) + (x + x)1z unit = xyz + xyz + x1z + x1z distributive = xyz + xyz + x(y + y)z + x(y + y)z unit = xyz + xyz + xyz + xyz + xyz + xyz distributive = xyz + xyz + xyz + xyz + xyz idempotent

7

Representing Boolean Functions

How to construct a Boolean expression that represents a Boolean Function?

F

F(x, y, z) =

x·y·z + x·y·z + x·y·z + x·y·z

1 0 0 0

1 0 1 0

1 1 0 0

1 1 1 1

x y z

(^0111) 0 1 0 1 0 0 1 0 0 0 0 1

8

Boolean Identities (again)

Double complement: x = x Idempotent laws: x +x = x, x ·x =x Identity laws: x + 0 =x, x · 1 =x Domination laws: x + 1 = 1 , x · 0 = 0 Commutative laws: x +y =y + x, x · y =y ·x

Associative laws: x + (y + z) = (x +y) + z x · (y ·z) = (x ·y) ·z Distributive laws: x +y·z = (x +y)·(x +z) x · (y +z) = x·y + x·z De Morgan’s laws: (x ·y) =x +y, (x +y) =x ·y Absorption laws: x +x·y = x, x · (x +y) =x

the Unit Property: x + x = 1 and Zero Property: x ·x = 0

10

Dual Expression (related to identity pairs)

The dual e d^ of a Boolean expression e is obtained

by exchanging + with • , and 0 with 1 in e.

Example: e = xy + zw e = x + y + 0

Duality principle: e 1 ⇔ e 2 iff e 1 d⇔ e 2 d

x (x + y) = x iff x + xy = x (absorption)

e d^ =(x + y)(z + w) e d^ = x · y · 1

11

Dual Function

The dual of a Boolean function F represented by a Boolean expression is the function represented by the dual of this expression.

The dual function of F is denoted by Fd

13

CNF: Conjunctive Normal Form

A literal is a Boolean variable or its complement.

A maxterm of Boolean variables x 1 ,…,xn is a

Boolean sum of n literals y 1 …yn , where yi is either

the literal xi or its complement xi.

Example:

(x +y + z) •^ ( x + y + z)^ •^ ( x + y +z)

Conjuctive Normal Form: product of sums

maxterms

14

Conjunctive Normal Form

To find the CNF representation of a Boolean

function F

1. Find the DNF representation of its complement F
2. Then complement both sides and apply DeMorgan’s laws to get F:

1 0 0 0

1 0 1 0

1 1 0 0

1 1 1 1

x y z

(^0111) 0 1 0 1 0 0 1 0 0 0 0 1

F

1

1

1

0

0 0 1 0

F (^) F = x·y·z + x·y·z + x·y·z + x·y·z

= (x+y+z) · (x+y+z) · (x+y+z) · (x+y+z)

F = (x·y·z) · (x·y·z) · (x·y·z) · (x·y·z)

16

Logic Gates: the basic elements of circuits

Electronic circuits consist of so-called gates connected by wires

OR gate

AND gate

x

y

xy

x (^) x Inverter (NOT gate)

x y

x+y

17

Multiway Logical Gates

Multiple Input AND, OR Gates

x (^1) x 1 • x 2 • … • x (^) n x (^2) x (^) n

x (^1) x 1 +x 2 +… + x (^) n x (^2) x (^) n

19

Circuit Design

Design a circuit for xy + xy

x y xy + xy

20

Circuit Design

Design a circuit for xy + xy 0 0 1

0 1 0

1 0 0

1 1 1

x y F

x y xy + xy