Mathematical Background

1. Mathematical Background Appendix A

2. Boolean Logic Wffs A wff is any string that is formed according to the following rules: • A propositional symbol (or variable) is a wff. • If P is a wff, then P is a wff. • If P and Q are wffs, then so are: PQ, PQ, PQ, and PQ. • If P is a wff, then (P) is a wff.

3. Truth Tables Define Operators

4. When Wffs are True • A Boolean wff is valid or is a tautology iff it is true for all assignments of truth values to the variables it contains. • A Boolean wff is satisfiable iff it is true for at least one assignment of truth values to the variables it contains. • A Boolean wff is unsatisfiable iff it is false for all assignments of truth values to the variables it contains. • Two wffs P and Q are equivalent, written PQ, iff they have the same truth values regardless of the truth values of the variables they contain.

5. Using Truth Tables PP is a tautology:

6. Properties of Boolean Operators •  and  are commutative and associative. •  is commutative but not associative. •  and  are idempotent: (e.g., (PP) P). •  and  distribute over each other: • P (QR)  (PQ)  (PR). • P (QR)  (PQ)  (PR).

7. More Properties • Absorption laws: • P (PQ) P. • P (PQ) P. • Double negation: P P. • de Morgan’s Laws: • (PQ)  (P Q). • (PQ)  (P Q).

8. Entailment A set A of wffs logically implies or entails a conclusion Q iff, whenever all of the wffs in A are true, Q is also true. Example: A B  C entail A  D D

9. Inference Rules • An inference rule is sound iff, whenever it is applied to a set A of axioms, any conclusion that it produces is entailed by A. An entire proof is sound iff it consists of a sequence of inference steps each of which was constructed using a sound inference rule. • A set of inference rules R is complete iff, given any set A of axioms, all statements that are entailed by A can be proved by applying the rules in R.

10. Some Sound Inference Rules • Modus ponens: From (PQ) and P, conclude Q. • Modus tollens: From (PQ) and Q, conclude P. • Or introduction: From P, conclude (PQ). • And introduction: From P and Q, conclude (PQ). • And elimination: From (PQ), conclude P or conclude Q.

11. First-Order Logic A well-formed formula (wff) in first-order logic is an expression that can be formed by: • If P is an n-ary predicate and each of the expressions x1, x2, … , xn is a term, then an expression of the form P(x1, x2, … , xn) is a wff. If any variable occurs in such a wff, then that variable is free. • If P is a wff, then P is a wff. • If P and Q are wffs, then so are PQ, PQ, PQ, and PQ. • If P is a wff, then (P) is a wff. • If P is a wff, then x (P) and x (P) are wffs. Any free instance of x in P is bound by the quantifier and is then no longer free.

12. Sentences A wff with no free variables is called a sentence or a statement. • Bear(Smoky). • x (Bear(x) Animal(x)). • x (Animal(x) Bear(x)). • x (Animal(x) y (Mother-of(y, x))). • x ((Animal(x) Dead(x)) Alive(x)). A ground instance is a sentence that contains no variables.

13. Truth • Bear(Smokey). • x (Bear(x) Animal(x)). • x (Animal(x) Bear(x)). • x (Animal(x) y (Mother-of(y, x))). • x ((Animal(x) Dead(x)) Alive(x)). Which of these are true in the everyday world?

14. Interpretations and Models • An interpretation for a sentence w is a pair (D, I), where D is a universe of objects. I assigns meaning to the symbols of w: it assigns values, drawn from D, to the constants in w and it assigns functions and predicates (whose domains and ranges are subsets of D) to the function and predicate symbols of w. • A model of a sentence w is an interpretation that makes w true. For example, let w be the sentence: x (y (y < x)). • A sentence w is valid iff it is true in all interpretations. • A sentence w is satisfiable iff there exists some interpretation in which w is true. • A sentence w is unsatisfiable iff w is valid.

15. Examples • x ((P(x) Q(Smoky)) P(x)). • (x (P(x) (P(x))). • x (P(x, x)).

16. Additional Sound Inference Rules • Quantifier exchange: • From x (P), conclude x (P). • From x (P), conclude x (P). • From x (P), conclude x (P). • From x (P), conclude x (P) . • Universal instantiation: For any constant C, from x (P(x)), conclude P(C). • Existential generalization: For any constant C, from P(C) conclude x (P(x)).

17. A Simple Proof Assume the following three axioms: [1] x (P(x) Q(x) R(x)). [2] P(X1). [3] Q(X1). We prove R(X1) as follows: [4] P(X1) Q(X1) R(X1). (Universal instantiation, [1].) [5] P(X1) Q(X1). (And introduction, [2], [3].) [6] R(X1). (Modus ponens, [5], [4].)

18. Gödel’s Theorems • Completeness Theorem: there exists some set of inference rules R such that, given any set of axioms A and a sentence c, there is a proof of c, starting with A and applying the rules in R, iff c is entailed by A. • Incompleteness Theorem: any theory that is derived from a decidable set of axioms and that characterizes the standard behavior of the constants 0 and 1, plus the functions plus and times on the natural numbers, cannot be both consistent and complete.

19. Sets • S1 = {13, 11, 8, 23}. • S2 = {8, 23, 11, 13}. • S3 = {8, 8, 23, 23, 11, 11, 13, 13}. • S4 = {apple, pear, banana, grape}. • S5 = {January, February, March, April, May, June, July, August, September, October, November, December}. • S6 = {x : xS5 and x has 31 days}. • S7 = {January, March, May, July, August, October, December}.

20. Sets • ℕ = the nonnegative integers (also called the natural numbers). • Z = the integers ( … -3, -2, -1, 0, 1, 2, 3, …).

21. Sets • S8 = {i : x ℕ (i = 2x)}. • S9 = {0, 2, 4, 6, 8, …}. • S10 = the even natural numbers. • S11 = the syntactically valid C programs. • S12 = {x : xS11 and x never gets into an infinite loop}. • S13 = {finite length strings of a’s and b’s}.

22. Defining a Set • Write a program that enumerates the elements of S. • Write a program that decidesS by implementing the characteristic function of S. Such a program returns True if run on some element that is in S and False if run on an element that is not in S.

23. Cardinality The cardinality of every set we will consider is: • a natural number (if S is finite), • “countably infinite” (if S has the same number of elements as there are integers), or • “uncountably infinite” (if S has more elements than there are integers).

24. Relating Sets to Each Other

25. Sets of Sets • The power set of A is the set of all subsets of A. Let A = {1, 2, 3}. Then: P(A) = {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. •  P(A) is a partition of a set A iff: • no element of  is empty, • all pairs of elements of  are disjoint , and • the union of all the elements of  equals A. Partitions of A: {{1}, {2, 3}} or {{1, 3}, {2}} or {{1, 2, 3}}.

26. What is a Relation? An ordered pair is a sequence of two objects, written: (x, y). Order matters. So (unless x and y are equal): (x, y)  (y, x).

27. Cartesian Products The Cartesian product of two sets A and B is the set of all ordered pairs (a, b) such that aA and b B. We write it as: AB If A and B are finite, the cardinality of their Cartesian product is: |AB| = |A||B|.

28. Cartesian Products Let A be: {Dave, Sara, Billy} Let B be: {cake, pie, ice cream} AB = { (Dave, cake), (Dave, pie), (Dave, ice cream), (Sara, cake), (Sara, pie), (Sara, ice cream), (Billy, cake), (Billy, pie), (Billy, ice cream)}. B A = { (cake, Dave), (pie, Dave), (ice cream, Dave), (cake, Sara), (pie, Sara), (ice cream, Sara), (cake, Billy), (pie, Billy), (ice cream, Billy)}.

29. Relations An n-ary relation over sets A1, A2, … An is a subset of: A1A2 … An. A binary relation over two sets A and B is a subset of: AB. Example: Dessert = { (Dave, cake), (Dave, ice cream), (Sara, pie), (Sara, ice cream)} Dessert -1 = { (cake, Dave), (ice cream, Dave), (pie, Sara), (ice cream, Sara)}

30. Composing Relations The composition of R1AB and R2B , written R2R1, is: R2R1 = {(a, c) : b ((a, b) R1 ((b, c) R2)} Example: Dessert = {(Dave, cake), (Dave, ice cream), (Sara, pie), (Sara, ice cream)} Fatgrams = {(cake, 30), (pie, 25), (ice cream, 15)} Dessert  Fatgrams = {(Dave, 30), (Dave, 15), (Sara, 25), (Sara, 15)}

31. Representing Relations Ways to represent a binary relation R: • List the elements of R. • Write a procedure that defines R either by: • Enumerating it. • Deciding it. • Encode R as an adjacency matrix. • Encode R as a directed graph.

32. Representing a Binary Relation as an Adjacency Matrix

33. Representing Binary Relations as Graphs

34. Properties of Relations • RAA is reflexive iff, x A ((x, x) R). • Examples: • Address defined as “lives at same address as”. •  defined on the integers. For every integer x, xx.

35. Properties of Relations • RAA is symmetric iff x, y ((x, y) R (y, x) R). • Examples: • Address is symmetric. •  is not symmetric.

36. Equivalence Relations • A relation RAA is an equivalence relation iff it is: • reflexive, • symmetric, and • transitive. • Examples: • Equality • Lives-at-Same-Address-As • Same-Length-As

37. Equivalence Classes Make it reflexive:

38. Equivalence Classes Add (1, 2):

39. Equivalence Classes Add (2, 3):

40. Equivalence Classes

41. Equivalence Classes An equivalence relation R on a set S carves S up into a set of clusters or islands, which we’ll call equivalence classes. This set of equivalence classes has the following key property: s, tS ((sclassi (s, t) R) tclassi). If R is an equivalence relation on a nonempty set A, then the set of equivalence classes of R is a partition  of A. Because  is a partition: (a) no element of  is empty; (b) all members of  are disjoint; and (c) the union of all the elements of  equals A.

42. Partial Orders • A partial order is a relation that is: • reflexive, • antisymmetric, and • transitive. • Let R be a partial order defined on a set A. Then the pair (A, R) is a partially ordered set.

43. Subset-of

44. A Concept Hierarchy

45. A Subsumption Lattice

46. Total Orders A total orderRAA is a partial order that has the additional property that: x, y A ((x, y) R (y, x) R). Example:  If R is a total order defined on a set A, then the pair (A, R) is a totally ordered set. 6 5 4 3

47. Well-Founded and Well-Ordered Sets • Given a partially ordered set (A, R), an infinite descending chain is a totally ordered, with respect to R, subset B of A that has no minimal element. • If (A, R) contains no infinite descending chains then it is called a well-founded set. • Used for halting proofs. • If (A, R) is a well-founded set and R is a total order, then (A, R) is called a well-ordered set. • Used in induction proofs.

48. Well-Founded and Well-Ordered Sets

49. Functions A functionf from a set A to a set B is a binary relation, subset of AB, such that: xA ((((x, y) f (x, z) f) y = z) yB ((x, y) f )). Dessert = { (Dave, cake), (Dave, ice cream), (Sara, pie), (Sara, ice cream)} is not a function. succ(n) = n + 1 is a function.

50. Properties of Functions • f : AB is a total function on A iff it is a function that is defined on all elements of A. • f : AB is a partial function on A iff f is a subset of AB and every element of A is related to no more than one element of B. • f : AB is one-to-one iff no two elements of A map to the same element of B. • f : AB is onto iff every element of B is the value of some element of A.