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Discrete Mathematics CS 2610. September 16, 2008. Monotonic Real Functions. Let f : A → B such that A,B R f 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:
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Discrete Mathematics CS 2610 September 16, 2008
Monotonic Real Functions • Let f: A→B such that A,B R • f 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
Increasing Functions are Injective Theorem: A strictly increasing function is always injective Proof: Let x, y be 2 numbers st x ≠ y, now show that their images are not equal. Either x > y, or y > x. 1) x > y: since f is strictly increasing then f(x) > f(y), i.e., f(x) ≠ f(y)! 2) y > x: since f is strictly increasing then f(y) > f(x), i.e., f(y) ≠ f(x)! x ≠ y implies f(x) ≠ f(y) so the function f is injective.
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”
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
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
Boolean Algebras • Boolean algebra provides the operations and the rules for working with the set {0, 1}. • The rules of Boolean algebra will be familiar. It is essentially the same as Boolean logic, except that we manipulate symbols within the context of an algebra. • These are the rules that underlie electronic and optical circuits, and the methods we will discuss are fundamental to VLSI design.
Boolean Algebra • The minimal Boolean algebra is the algebra formed over the set of truth values {0, 1} by using the operations: +, •, ¯ (sum, product, 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
Boolean Algebra Tables _ x x x y x+y xy 1 0 1 1 1 1 0 1 1 0 1 0 0 1 1 0 0 0 0 0 x,y are Boolean variables – they assume values 0 or 1
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 n Z+, any function f :Bn→B is called a Boolean function of degree n.
Example Boolean Function • F(x,y,z) = B3→B x y z F(x,y,z)=x(y+z) 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0
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 • Let F and G be two Booleans functions of degree n. _ _______ • Complement of F: 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 • 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 • 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 E2 are 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 = E6E4 = -z( x + y) _ • Note: equivalent notation: -E = E for complement
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. _ _ x1 x2 x3 F(x1,x2,x3)=x1(x2+x3)+x1x2x3 1 1 1 1 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 0 0 0 0 1
Boolean Functions • Two Boolean expressions E1 and E2 that represent the exact same function F are called equivalent x1 x2 x3 F(x1,x2,x3) 1 1 1 1 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 0 0 0 0 1 _ _ F(x1,x2,x3) = x1(x2+x3)+x1x2x3 _ _ _ F(x1,x2,x3) = x1x2+x1x3+ x1x2x3
Representing Boolean Functions • How to construct a Boolean expression that represents a Boolean Function? • Create the “sum of products” expansion. • Then you can simplify using 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 Boolean Identities the Unit Property: x + x = 1 and Zero Property: x · x = 0
