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Discrete Mathematics CS 2610. February 12, 2009. Agenda. Previously Finished functions Began Boolean algebras And now Continue with Boolean algebras. But First. p  q  r, is NOT true when only one of p, q, or r is true. Why not? It is true for (p Λ ¬q Λ ¬r)

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Presentation Transcript
agenda
Agenda
  • Previously
    • Finished functions
    • Began Boolean algebras
  • And now
    • Continue with Boolean algebras
but first
But First
  • p  q  r, is NOT true when only one of p, q, or r is true. Why not?
  • It is true for (p Λ ¬q Λ ¬r)
  • It is true for (¬p Λ q Λ ¬r)
  • It is true for (¬p Λ ¬q Λ r)
  • So what’s wrong? Raise your hand when you know.
injective functions one to one
Injective Functions (one-to-one)
  • If function f : A  B is 1-to-1 then every b  B has 0 or 1 pre-image.
  • Proof (bwoc): Say f is 1-to-1 and b  B has 2 or more pre-images.
  • Then a1, a2 st a1  A and a2  A, and a1 ≠ a2.
  • So f(a1) = b and f(a2) = b, meaning f(a1) = f(a2).
  • This contradicts the definition of an injection since when a1 ≠ a2 we know f(a1) ≠ f(a2).
boolean algebras chapter 11
Boolean Algebras (Chapter 11)
  • Boolean algebra provides the operations and the rules for working with the set {0, 1}.
  • These are the rules that underlie electronic and optical circuits, and the methods we will discuss are fundamental to VLSI design.
boolean algebra
Boolean Algebra
  • The minimal Boolean algebra is the algebra formed over the set of truth values {0, 1} by using the operations functions +, ·, - (sum, product, complement).
  • The minimal Boolean algebra is equivalent to propositional logic where
    • Ocorresponds to False
    • 1 corresponds to True
    •  corresponds logical operator AND
    • + corresponds logical operator OR
    • - corresponds logical operator NOT
boolean algebra tables

x

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x + y

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Boolean Algebra Tables

x,y are Boolean variables – they assume values 0 or 1

boolean n tuples
Boolean n-Tuples
  • Let B = {0, 1}, the set of Boolean values.
  • LetBn = { (x1,x2,…xn) | xi B, i=1,..,n}

.

B1= { (x1) | x1 B,}

B2= { (x1, x2), | xi B, i=1,2}

Bn= { ((x1,x2,…xn) | xi B, i=1,..,n,}

  • For all nZ+, any function f:Bn→B is called a Boolean function of degree n.
example boolean function
Example Boolean Function

F(x,y,z) =B3B

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F(x,y,z)=x(y+z)

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B3 has 8 triplets

1

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number of boolean functions
Number of Boolean Functions
  • How many different Boolean functions of degree 1 are there?
  • How many different Boolean functions of degree 2 are there?
  • How many different functions of degree n are there ?
    • There are 22ⁿ distinct Boolean functions of degree n.
combining boolean functions
Combining Boolean Functions
  • Let Fand G be two Booleans functions of degree n.
    • Complement ofF: F (x1,..xn) = F (x1,..xn)
    • Boolean Sum : (F + G)(x1,..xn) = F (x1,..xn) + G (x1,..xn)
    • Boolean Product: (F·G)(x1,..xn) = F(x1,..xn)·G(x1,..xn)
equal boolean functions
Equal Boolean Functions
  • Two Boolean functions F and G of degree n are equal iff for all (x1,..xn)  Bn, F(x1,..xn) = G(x1,..xn)
  • Example: F(x,y,z) = x(y+z), G(x,y,z) = xy + zx
boolean expressions
Boolean Expressions
  • Let x1, …, xn be n different Boolean variables.
  • A Boolean expression is a string of one of the following forms (recursive definition):
    • 0, 1, x1, …, or xn. are Boolean Expressions
    • If E1 and E2are Boolean expressions then -E1, (E1E2), or (E1+E2)are Boolean expressions.

Example:

E1 = x

E2 = y

E3 = z

E4 = E1 + E2= x + y

E5 = E1E2= x y

E6 = -E3 = -z

E7 = E6 + E4 = -z + x + y

E8 = E6 E4 = -z ( x + y)

Note: equivalent notation: -E = E for complement

functions and expressions

x1

x2

x3

F(x1,x2,x3)

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F(x1,x2,x3) = x1(x2+x3)+x1x2x3

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Functions and Expressions
  • A Boolean expression represents a Boolean function.
    • Furthermore, every Boolean function (of a given degree) can be represented by a Boolean expression with n variables.
boolean functions

x1

x2

x3

F(x1,x2,x3)

0

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F(x1,x2,x3) = x1(x2+x3)+x1x2x3

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F(x1,x2,x3) = x1x2+x1x3+x1x2x3

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Boolean Functions
  • Two Boolean expressions e1and e2 that represent the exact same function F are called equivalent
representing boolean functions

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Representing Boolean Functions
  • How to construct a Boolean expression that represents a Boolean Function ?

F

F(x, y, z) = 1 if and only if:

(-x)(y)(-z) + (-x)yz + x(-y)z + xyz

What about a 2-input multiplexer?

boolean identities
Boolean Identities
  • 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

boolean identities1
Boolean Identities
  • Absorption law:
    • Show that x ·(x + y)=x
    • x ·(x + y)= (x + 0) ·(x + y) identity
    • = x + 0 ·y distributive *
    • = x + y · 0 commutative
    • = x + 0 domination
    • = x identity