Discrete Mathematics CS 2610

February 10, 2009

2

Agenda

Previously Functions

And now Finish functions Start Boolean algebras (Sec. 11.1)

3

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.

4

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).

5

Combining Real Functions

Given f :RR and g :RR then

(f g): RR, is defined as

(f g)(x) = f(x) g(x)

(f · g): RR is defined as

(f · g)(x) = f(x) · g(x)

Example:

Let f :RR be f(x) = 2x and and g :RR be g(x) = x3

(f+g)(x) = x3+2x

(f · g)(x) = 2x4

6

Monotonic Real Functions

Let f: AB such that A,B Rf is strictly increasing iff

for all x, y A x > y f(x) > f(y)

f is strictly decreasing iff for all x, y A, x > y f(x) < f(y)

Example:

f: R+ R+, f(x) = x2 is strictly increasing

7

Increasing Functions are Injective

Theorem: A strictly increasing function is always injective

Proof:

8

Floor and Ceiling Function

Definition: The floor function .:R→Z, x is the largest integer which is less than or equal to x.

x reads the floor of x

Definition: The ceiling function . :R→Z, x is the smallest integer which is greater than or equal to x.

x reads the ceiling of x

9

Example Ceiling and Floor Functions

Example:

-2.8 =

2.8 =

2.8 =

-2.8 =

-3

2

3

-2

10

Ceiling and Floor Properties

Let n be an integer

(1a) x = n if and only if n ≤ x < n+1

(1b) x = n if and only if n-1 < x ≤ n

(1c) x = n if and only if x-1 < n ≤ x

(1d) x = n if and only if x ≤ n < x+1

(2) x-1 < x ≤ x ≤ x < x+1

(3a) -x = - x

(3b) -x = - x

(4a) x+n = x+n

(4b) x+n = x+n

11

Ceiling and Floor Functions

Let n be an integer, prove x+n = x+n

Proof Let k = x Then k ≤ x < k+1 So k+n ≤ x+n < k+1+n I.e., k+n ≤ x+n < (k+n)+1 Since both k and n are integers, k+n is an

integer Thus, x+n = k+n = x+n (by our choice of k)

This concludes the proof This also concludes Chapter 2!

12

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.

13

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, and complement).

The minimal Boolean algebra is equivalent to propositional logic where O corresponds to False 1 corresponds to True corresponds logical operator AND + corresponds logical operator OR - corresponds logical operator NOT

14

Boolean Algebra Tables

x

0

0

1

1

y

0

1

0

1

x + y

0

1

1

1

xy

0

0

0

1

x

0

1

x

1

0

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

15

Boolean n-Tuples

Let B = {0, 1}, the set of Boolean values.

Let Bn = { (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 nZ+, any function f:Bn→B is called a Boolean function of degree n.

16

Example Boolean Function

x

0

0

0

0

1

1

1

1

y

0

0

1

1

0

0

1

1

z

0

1

0

1

0

1

0

1

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

F(x,y,z) =B3B

B3 has 8 triplets

0

0

0

0

1

1

0

1

17

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.