… and now for something completely different…

… and now for something completely different…
Set Theory Actually, you will see that logic and set theory are very closely related. Spring 2003 CMSC Discrete Structures

CMSC 203 - Discrete Structures
Set Theory Set: Collection of objects (called elements) aA “a is an element of A” “a is a member of A” aA “a is not an element of A” A = {a1, a2, …, an} “A contains a1, …, an” Order of elements is insignificant It does not matter how often the same element is listed (repetition doesn’t count). Spring 2003 CMSC Discrete Structures

Set Equality Sets A and B are equal if and only if they contain exactly the same elements. Examples: A = {9, 2, 7, -3}, B = {7, 9, -3, 2} : A = B A = {dog, cat, horse}, B = {cat, horse, squirrel, dog} : A  B A = {dog, cat, horse}, B = {cat, horse, dog, dog} : A = B Spring 2003 CMSC Discrete Structures

Examples for Sets “Standard” Sets: Natural numbers N = {0, 1, 2, 3, …} Integers Z = {…, -2, -1, 0, 1, 2, …} Positive Integers Z+ = {1, 2, 3, 4, …} Real Numbers R = {47.3, -12, , …} Rational Numbers Q = {1.5, 2.6, -3.8, 15, …} (correct definitions will follow) Spring 2003 CMSC Discrete Structures

Examples for Sets A =  “empty set/null set” A = {z} Note: zA, but z  {z} A = {{b, c}, {c, x, d}} set of sets A = {{x, y}} Note: {x, y} A, but {x, y}  {{x, y}} A = {x | P(x)} “set of all x such that P(x)” P(x) is the membership function of set A x (P(x)  xA) A = {x | x N  x > 7} = {8, 9, 10, …} “set builder notation” Spring 2003 CMSC Discrete Structures

Examples for Sets We are now able to define the set of rational numbers Q: Q = {a/b | aZ  bZ+}, or Q = {a/b | aZ  bZ  b0} And how about the set of real numbers R? R = {r | r is a real number} That is the best we can do. It can neither be defined by enumeration nor builder function. Spring 2003 CMSC Discrete Structures

Subsets A  B “A is a subset of B” A  B if and only if every element of A is also an element of B. We can completely formalize this: A  B  x (xA xB) Examples: A = {3, 9}, B = {5, 9, 1, 3}, A  B ? true A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A  B ? true A = {1, 2, 3}, B = {2, 3, 4}, A  B ? false Spring 2003 CMSC Discrete Structures

Subsets Useful rules: A = B  (A  B)  (B A) (A  B) (B  C)  A  C (see Venn Diagram) U C B A Spring 2003 CMSC Discrete Structures

Subsets Useful rules:   A for any set A (but   A may not hold for any set A) A  A for any set A Proper subsets: A  B “A is a proper subset of B” A  B  x (xA  xB)  x (xB  xA) or A  B  x (xA  xB)  x (xB xA) Spring 2003 CMSC Discrete Structures

Cardinality of Sets If a set S contains n distinct elements, nN, we call S a finite set with cardinality n. Examples: A = {Mercedes, BMW, Porsche}, |A| = 3 B = {1, {2, 3}, {4, 5}, 6} |B| = 4 C =  |C| = 0 D = { xN | x  7000 } |D| = 7001 E = { xN | x  7000 } E is infinite! Spring 2003 CMSC Discrete Structures

The Power Set P(A) “power set of A” (also written as 2A) P(A) = {B | B  A} (contains all subsets of A) Examples: A = {x, y, z} P(A) = {, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}} A =  P(A) = {} Note: |A| = 0, |P(A)| = 1 Spring 2003 CMSC Discrete Structures

The Power Set Cardinality of power sets: | P(A) | = 2|A| Imagine each element in A has an “on/off” switch Each possible switch configuration in A corresponds to one subset of A, thus one element in P(A) z y x 8 7 6 5 4 3 2 1 A For 3 elements in A, there are 222 = 8 elements in P(A) Spring 2003 CMSC Discrete Structures

Cartesian Product The ordered n-tuple (a1, a2, a3, …, an) is an ordered collection of n objects. Two ordered n-tuples (a1, a2, a3, …, an) and (b1, b2, b3, …, bn) are equal if and only if they contain exactly the same elements in the same order, i.e. ai = bi for 1  i  n. The Cartesian product of two sets is defined as: AB = {(a, b) | aA  bB} Spring 2003 CMSC Discrete Structures

Cartesian Product Example: A = {good, bad}, B = {student, prof} AB = { (good, student), (good, prof), (bad, student), (bad, prof)} (prof, bad)} (student, good), (prof, good), (student, bad), BA = { Example: A = {x, y}, B = {a, b, c} AB = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c)} Spring 2003 CMSC Discrete Structures

Cartesian Product Note that: A =  A =  For non-empty sets A and B: AB  AB  BA |AB| = |A||B| The Cartesian product of two or more sets is defined as: A1A2…An = {(a1, a2, …, an) | aiAi for 1  i  n} Spring 2003 CMSC Discrete Structures

Set Operations Union: AB = {x | xA  xB} Example: A = {a, b}, B = {b, c, d} AB = {a, b, c, d} Intersection: AB = {x | xA  xB} AB = {b} Cardinality: |AB| = |A| + |B| - |AB| Spring 2003 CMSC Discrete Structures

Set Operations Two sets are called disjoint if their intersection is empty, that is, they share no elements: AB =  The difference between two sets A and B contains exactly those elements of A that are not in B: A-B = {x | xA  xB} Example: A = {a, b}, B = {b, c, d}, A-B = {a} Cardinality: |A-B| = |A| - |AB| Spring 2003 CMSC Discrete Structures

Set Operations The complement of a set A contains exactly those elements under consideration that are not in A: denoted Ac (or as in the text) Ac = U-A Example: U = N, B = {250, 251, 252, …} Bc = {0, 1, 2, …, 248, 249} Spring 2003 CMSC Discrete Structures

Logical Equivalence Equivalence laws Identity laws, P  T  P, Domination laws, P  F  F, Idempotent laws, P  P  P, Double negation law,  ( P)  P Commutative laws, P  Q  Q  P, Associative laws, P  (Q  R) (P  Q)  R, Distributive laws, P  (Q  R) (P  Q)  (P  R), De Morgan’s laws,  (PQ)  ( P)  ( Q) Law with implication P  Q   P  Q Spring 2003 CMSC Discrete Structures

Set Identity Table 1 in Section 1.7 shows many useful equations Identity laws, A = A, AU = A Domination laws, AU = U, A =  Idempotent laws, AA = A, AA = A Complementation law, (Ac)c = A Commutative laws, AB = BA, AB = BA Associative laws, A(B  C) = (AB)C, … Distributive laws, A(BC) = (AB)(AC), … De Morgan’s laws, (AB)c = AcBc, (AB)c = AcBc Absorption laws, A(AB) = A, A(AB) = A Complement laws, AAc = U, AAc =  Spring 2003 CMSC Discrete Structures

Set Identity How can we prove A(BC) = (AB)(AC)? Method I: logical equivalent xA(BC) xA  x(BC) xA  (xB  xC) (xA  xB)  (xA  xC) (distributive law) x(AB)  x(AC) x(AB)(AC) Every logical expression can be transformed into an equivalent expression in set theory and vice versa. Spring 2003 CMSC Discrete Structures

Set Operations Method II: Membership table 1 means “x is an element of this set” 0 means “x is not an element of this set” 1 (AB) (AC) AC AB A(BC) BC A B C Spring 2003 CMSC Discrete Structures

… and the following mathematical appetizer is about…
Functions Spring 2003 CMSC Discrete Structures

Functions A function f from a set A to a set B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f: AB (note: Here, ““ has nothing to do with if… then) Spring 2003 CMSC Discrete Structures

Functions If f:AB, we say that A is the domain of f and B is the codomain of f. If f(a) = b, we say that b is the image of a and a is the pre-image of b. The range of f:AB is the set of all images of all elements of A. We say that f:AB maps A to B. Spring 2003 CMSC Discrete Structures

Functions Let us take a look at the function f:PC with P = {Linda, Max, Kathy, Peter} C = {Boston, New York, Hong Kong, Moscow} f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = New York Here, the range of f is C. Spring 2003 CMSC Discrete Structures

Functions Let us re-specify f as follows: f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = Boston Is f still a function? yes What is its range? {Moscow, Boston, Hong Kong} Spring 2003 CMSC Discrete Structures

Functions Other ways to represent f: Boston Peter Hong Kong Kathy Max Moscow Linda f(x) x Linda Max Kathy Peter Boston New York Hong Kong Moscow Spring 2003 CMSC Discrete Structures

Functions If the domain of our function f is large, it is convenient to specify f with a formula, e.g.: f:RR f(x) = 2x This leads to: f(1) = 2 f(3) = 6 f(-3) = -6 … Spring 2003 CMSC Discrete Structures

Functions Let f1 and f2 be functions from A to R. Then the sum and the product of f1 and f2 are also functions from A to R defined by: (f1 + f2)(x) = f1(x) + f2(x) (f1f2)(x) = f1(x) f2(x) Example: f1(x) = 3x, f2(x) = x + 5 (f1 + f2)(x) = f1(x) + f2(x) = 3x + x + 5 = 4x + 5 (f1f2)(x) = f1(x) f2(x) = 3x (x + 5) = 3x2 + 15x Spring 2003 CMSC Discrete Structures

Functions We already know that the range of a function f:AB is the set of all images of elements aA. If we only regard a subset SA, the set of all images of elements sS is called the image of S. We denote the image of S by f(S): f(S) = {f(s) | sS} Spring 2003 CMSC Discrete Structures

Functions Let us look at the following well-known function: f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = Boston What is the image of S = {Linda, Max} ? f(S) = {Moscow, Boston} What is the image of S = {Max, Peter} ? f(S) = {Boston} Spring 2003 CMSC Discrete Structures

Properties of Functions
A function f:AB is said to be one-to-one (or injective), if and only if x, yA (f(x) = f(y)  x = y) In other words: f is one-to-one if and only if it does not map two distinct elements of A onto the same element of B. Spring 2003 CMSC Discrete Structures

And again… f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = Boston Is f one-to-one? No, Max and Peter are mapped onto the same element of the image. g(Linda) = Moscow g(Max) = Boston g(Kathy) = Hong Kong g(Peter) = New York Is g one-to-one? Yes, each element is assigned a unique element of the image. Spring 2003 CMSC Discrete Structures

How can we prove that a function f is one-to-one? Whenever you want to prove something, first take a look at the relevant definition(s): x, yA (f(x) = f(y)  x = y) Example: f:RR f(x) = x2 Disproof by counterexample: f(3) = f(-3), but 3  -3, so f is not one-to-one. Spring 2003 CMSC Discrete Structures

… and yet another example: f:RR f(x) = 3x One-to-one: x, yA (f(x) = f(y)  x = y) To show: f(x)  f(y) whenever x  y (indirect proof) x  y 3x  3y f(x)  f(y), so if x  y, then f(x)  f(y), that is, f is one-to-one. Spring 2003 CMSC Discrete Structures

A function f:AB with A,B  R is called strictly increasing, if x,yA (x < y  f(x) < f(y)), and strictly decreasing, if x,yA (x < y  f(x) > f(y)). Obviously, a function that is either strictly increasing or strictly decreasing is one-to-one. Spring 2003 CMSC Discrete Structures

A function f:AB is called onto, or surjective, if and only if for every element bB there is an element aA with f(a) = b. In other words, f is onto if and only if its range is its entire codomain. A function f: AB is a one-to-one correspondence, or a bijection, if and only if it is both one-to-one and onto. Obviously, if f is a bijection and A and B are finite sets, then |A| = |B|. Spring 2003 CMSC Discrete Structures

Examples: In the following examples, we use the arrow representation to illustrate functions f:AB. In each example, the complete sets A and B are shown. Spring 2003 CMSC Discrete Structures

Linda Max Kathy Peter Boston New York Hong Kong Moscow Is f injective? No. Is f surjective? Is f bijective? Spring 2003 CMSC Discrete Structures

Linda Max Kathy Peter Boston New York Hong Kong Moscow Is f injective? No. Is f surjective? Yes. Is f bijective? Paul Spring 2003 CMSC Discrete Structures

Linda Max Kathy Peter Boston New York Hong Kong Moscow Lübeck Is f injective? Yes. Is f surjective? No. Is f bijective? Spring 2003 CMSC Discrete Structures

Linda Max Kathy Peter Boston New York Hong Kong Moscow Lübeck Is f injective? No! f is not even a function! Spring 2003 CMSC Discrete Structures

Linda Boston Is f injective? Yes. Is f surjective? Is f bijective? Max New York Kathy Hong Kong Peter Moscow Helena Lübeck Spring 2003 CMSC Discrete Structures

Inversion An interesting property of bijections is that they have an inverse function. The inverse function of the bijection f:AB is the function f-1:BA with f-1(b) = a whenever f(a) = b. Spring 2003 CMSC Discrete Structures

Inversion Example: f(Linda) = Moscow f(Max) = Boston f(Kathy) = Hong Kong f(Peter) = Lübeck f(Helena) = New York Clearly, f is bijective. The inverse function f-1 is given by: f-1(Moscow) = Linda f-1(Boston) = Max f-1(Hong Kong) = Kathy f-1(Lübeck) = Peter f-1(New York) = Helena Inversion is only possible for bijections (= invertible functions) Spring 2003 CMSC Discrete Structures

Inversion Linda Boston f Max New York f-1 f-1:CP is no function, because it is not defined for all elements of C and assigns two images to the pre-image New York. Kathy Hong Kong Peter Moscow Helena Lübeck Spring 2003 CMSC Discrete Structures

Composition The composition of two functions g:AB and f:BC, denoted by fg, is defined by (fg)(a) = f(g(a)) This means that first, function g is applied to element aA, mapping it onto an element of B, then, function f is applied to this element of B, mapping it onto an element of C. Therefore, the composite function maps from A to C. Spring 2003 CMSC Discrete Structures

Composition Example: f(x) = 7x – 4, g(x) = 3x, f:RR, g:RR (fg)(5) = f(g(5)) = f(15) = 105 – 4 = 101 (fg)(x) = f(g(x)) = f(3x) = 21x - 4 Spring 2003 CMSC Discrete Structures

Composition Composition of a function and its inverse: (f-1f)(x) = f-1(f(x)) = x The composition of a function and its inverse is the identity function i(x) = x. Spring 2003 CMSC Discrete Structures

Graphs The graph of a function f:AB is the set of ordered pairs {(a, b) | aA and f(a) = b}. The graph is a subset of AB that can be used to visualize f in a two-dimensional coordinate system. Spring 2003 CMSC Discrete Structures

Floor and Ceiling Functions
The floor and ceiling functions map the real numbers onto the integers (RZ). The floor function assigns to rR the largest zZ with z  r, denoted by r. Examples: 2.3 = 2, 2 = 2, 0.5 = 0, -3.5 = -4 The ceiling function assigns to rR the smallest zZ with z  r, denoted by r. Examples: 2.3 = 3, 2 = 2, 0.5 = 1, -3.5 = -3 Spring 2003 CMSC Discrete Structures

Now, something about BooleanAlgebra (section 10.1) Spring 2003 CMSC Discrete Structures

Boolean Algebra Boolean algebra provides the operations and the rules for working with the set {0, 1}. These are the rules that underlie electronic circuits, and the methods we will discuss are fundamental to VLSI design. We are going to focus on three operations: Boolean complementation, Boolean sum, and Boolean product Spring 2003 CMSC Discrete Structures

Boolean Operations The complement is denoted by a bar (on the slides, we will use a minus sign). It is defined by -0 = 1 and -1 = 0. The Boolean sum, denoted by + or by OR, has the following values: 1 + 1 = 1, = 1, = 1, = 0 The Boolean product, denoted by  or by AND, has the following values: 1  1 = 1, 1  0 = 0, 0  1 = 0, 0  0 = 0 Spring 2003 CMSC Discrete Structures

Boolean Functions and Expressions
Definition: Let B = {0, 1}. The variable x is called a Boolean variable if it assumes values only from B. A function from Bn, the set {(x1, x2, …, xn) |xiB, 1  i  n}, to B is called a Boolean function of degree n. Boolean functions can be represented using expressions made up from the variables and Boolean operations. Spring 2003 CMSC Discrete Structures

The Boolean expressions in the variables x1, x2, …, xn are defined recursively as follows: 0, 1, x1, x2, …, xn are Boolean expressions. If E1 and E2 are Boolean expressions, then (-E1), (E1E2), and (E1 + E2) are Boolean expressions. Each Boolean expression represents a Boolean function. The values of this function are obtained by substituting 0 and 1 for the variables in the expression. Spring 2003 CMSC Discrete Structures

For example, we can create Boolean expression in the variables x, y, and z using the “building blocks” 0, 1, x, y, and z, and the construction rules: Since x and y are Boolean expressions, so is xy. Since z is a Boolean expression, so is (-z). Since xy and (-z) are expressions, so is xy + (-z). … and so on… Spring 2003 CMSC Discrete Structures

Example: Give a Boolean expression for the Boolean function F(x, y) as defined by the following table: 1 F(x, y) y x Possible solution: F(x, y) = (-x)y Spring 2003 CMSC Discrete Structures

Another Example: 1 F(x, y, z) z y x Possible solution I: F(x, y, z) = -(xz + y) Possible solution II: F(x, y, z) = (-(xz))(-y) Spring 2003 CMSC Discrete Structures

There is a simple method for deriving a Boolean expression for a function that is defined by a table. This method is based on minterms. Definition: A literal is a Boolean variable or its complement. A minterm of the Boolean variables x1, x2, …, xn is a Boolean product y1y2…yn, where yi = xi or yi = -xi. Hence, a minterm is a product of n literals, with one literal for each variable. Spring 2003 CMSC Discrete Structures

Consider F(x,y,z) again: F(x, y, z) = 1 if and only if: x = y = z = 0 or x = y = 0, z = 1 or x = 1, y = z = 0 Therefore, F(x, y, z) = (-x)(-y)(-z) + (-x)(-y)z + x(-y)(-z) 1 F(x, y, z) z y x Spring 2003 CMSC Discrete Structures

Definition: The Boolean functions F and G of n variables are equal if and only if F(b1, b2, …, bn) = G(b1, b2, …, bn) whenever b1, b2, …, bn belong to B. Two different Boolean expressions that represent the same function are called equivalent. For example, the Boolean expressions xy, xy + 0, and xy1 are equivalent. Spring 2003 CMSC Discrete Structures

The complement of the Boolean function F is the function –F, where –F(b1, b2, …, bn) = -(F(b1, b2, …, bn)). Let F and G be Boolean functions of degree n. The Boolean sum F+G and Boolean product FG are then defined by (F + G)(b1, b2, …, bn) = F(b1, b2, …, bn) + G(b1, b2, …, bn) (FG)(b1, b2, …, bn) = F(b1, b2, …, bn) G(b1, b2, …, bn) Spring 2003 CMSC Discrete Structures

Question: How many different Boolean functions of degree 1 are there? Solution: There are four of them, F1, F2, F3, and F4: 1 F3 F2 F4 F1 x Spring 2003 CMSC Discrete Structures

Question: How many different Boolean functions of degree 2 are there? Solution: There are 16 of them, F1, F2, …, F16: 1 F2 F1 F3 y x F8 F7 F9 F5 F4 F6 F11 F10 F12 F14 F13 F15 F16 Spring 2003 CMSC Discrete Structures

Question: How many different Boolean functions of degree n are there? Solution: There are 2n different n-tuples of 0s and 1s. A Boolean function is an assignment of 0 or 1 to each of these 2n different n-tuples. Therefore, there are 22n different Boolean functions. Spring 2003 CMSC Discrete Structures

Duality There are useful identities of Boolean expressions that can help us to transform an expression A into an equivalent expression B (see Table 5 on page 705 in the textbook). We can derive additional identities with the help of the dual of a Boolean expression. The dual of a Boolean expression is obtained by interchanging Boolean sums and Boolean products and interchanging 0s and 1s. Spring 2003 CMSC Discrete Structures

Duality Examples: The dual of x(y + z) is x + yz. The dual of -x1 + (-y + z) is (-x + 0)((-y)z). The dual of a Boolean function F represented by a Boolean expression is the function represented by the dual of this expression. This dual function, denoted by Fd, does not depend on the particular Boolean expression used to represent F. Spring 2003 CMSC Discrete Structures

Duality Therefore, an identity between functions represented by Boolean expressions remains valid when the duals of both sides of the identity are taken. We can use this fact, called the duality principle, to derive new identities. For example, consider the absorption law x(x + y) = x. By taking the duals of both sides of this identity, we obtain the equation x + xy = x, which is also an identity (and also called an absorption law). Spring 2003 CMSC Discrete Structures

Definition of a Boolean Algebra
All the properties of Boolean functions and expressions that we have discovered also apply to other mathematical structures such as propositions and sets and the operations defined on them. If we can show that a particular structure is a Boolean algebra, then we know that all results established about Boolean algebras apply to this structure. For this purpose, we need an abstract definition of a Boolean algebra. Spring 2003 CMSC Discrete Structures

Definition of a Boolean Algebra
Definition: A Boolean algebra is a set B with two binary operations  and , elements 0 and 1, and a unary operation – such that the following properties hold for all x, y, and z in B: x  0 = x and x  1 = x (identity laws) x  (-x) = 1 and x  (-x) = 0 (domination laws) (x  y)  z = x  (y  z) and (x  y)  z = x  (y  z) and (associative laws) x  y = y  x and x  y = y  x (commutative laws) x  (y  z) = (x  y)  (x  z) and x  (y  z) = (x  y)  (x  z) (distributive laws) Spring 2003 CMSC Discrete Structures

Logic Gates Electronic circuits consist of so-called gates. There are three basic types of gates: x -x inverter x y x+y OR gate x y xy AND gate Spring 2003 CMSC Discrete Structures

Logic Gates Example: How can we build a circuit that computes the function xy + (-x)y ? xy + (-x)y x y xy -x (-x)y Spring 2003 CMSC Discrete Structures

Logic, Sets, and Boolean Algebra
Logic Set Boolean Algebra False  0 True U 1 AB AB AB AB AB A+B A AC Compare the equivalence laws of them Spring 2003 CMSC Discrete Structures

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