Chapter 1. Fundamentals

1. Chapter 1. Fundamentals Weiqi Luo (骆伟祺) School of Software Sun Yat-Sen University Email：weiqi.luo@yahoo.com Office：A309

2. Textbook: B. Kolman, R.C. Busby & S.C. Ross, Discrete Mathematical Structures (Sixth Edition), Higher Education Press,2010.11. • References: 1. 屈婉玲，耿素云，张立昂，离散数学， 清华大学出版社 2 K.H. Rosen, 离散数学及其应用（英文版）（第6版）机械工业出版社

3. Chapter one: Fundamentals • 1.1. Sets and Subsets • 1.2. Operations on Sets • 1.3. Sequences • 1.4. Properties of Integers • 1.5. Matrices • 1.6. Mathematical Structures

4. 1.1. Sets and Subset • What is a set? A Set is any well-defined collection of objects called the elements or members of the set. Well-defined means that it is possible to describe if a given object belongs to the collection or not. • Describing a Set Way one: List the elements of the set between braces (finite elements) e.g. the set of all positive integers that are less than 4 : {1, 2, 3} Way two: Specify a property that the elements of the set have in common e.g. R={x | x is a real number } Property of the elements

5. 1.1. Sets and Subset • The order of the Set {1, 2, 3}={1, 3, 2}={2, 3, 1}={2, 1,3 }={3, 1, 2}={3, 2, 1} • Repeated elements can be ignored {1, 2, 3, 1} = {1, 2, 3} • Several commonly used sets Please refer to Example 3 in Page 2.

6. 1.1. Sets and Subset • The relationships between Element & Set Usually, we use uppercase letters such as A, B and C to denote sets, and lowercase letters such as a, b, c, x, y and z to denote the elements of sets Binary cases: for a given element x and set A 1: x belongs to A denoted by x ∈A 2: x does not belong to A denoted by x ∉ A Fuzzy Sets The collections of rich people, young girls, so on and so forth Note: The words rich, young, beautiful, cool, hot, fat, thin etc. are fuzzy (not well defined). Refer to Wikipedia for more details about Fuzzy mathematics: http://en.wikipedia.org/wiki/Fuzzy_mathematics

7. 1.1. Sets and Subset • Subset If every element of A is also an element of B, namely, if whether x ∈A then x ∈B, we say that A is a subset of B, denoted by A ⊆ B . Otherwise, . • Venn diagrams A ⊆ B B A A A B B

8. 1.1. Sets and Subset A B U A A=B: A ⊆ B & B ⊆ A An universal set (U) is a set containing all objects for which the discussion is meaningful.

9. 1.1. Sets and Subset • Example 10 Let A be a set and let B = {A, {A}}, then, since A and {A} are elements of B, we have A ∈ B and {A} ∈B. It follows that {A} ⊆ B and {{A}} ⊆ B. However, it is not true that A ⊆ B Why?

10. 1.1. Sets and Subset • The cardinality of a finite set A set A is called finite if it has ndistinct elements, where n∈N. In this case, n is the cardinality of A and is denoted by |A|. e.g. A={1,2,3,1} |A| = 3 B={a, b, c, d, e, a}, |B| = 5 |A| < |B| A set that is not finite is called infinite, for instances, N, Z, Q, R as mentioned in Example 3. the cardinality of infinite? Continuum hypothesis (the 1stHilbert's Problems): http://en.wikipedia.org/wiki/Continuum_hypothesis

11. 1.1. Sets and Subset • Power set of a set A If A is a set, then the set of all subsets of A is called the power set of A and is denoted by P(A). e.g. A={1,2,3} Then P(A) consists of the following subsets of A: {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, and {1,2,3} |P(A)| = 2^n, why? Assuming n = |A| ∈N

12. 1.1. Sets and Subset • Homework ex.5, ex.13, ex.14, ex.23

13. 1.2. Operations on Sets • Union If A and B are sets, we define their union as the set consisting of all elements that belong to A or B and denote it by A U B. A U B = { x | x ∈ A or x ∈ B } U A B

14. 1.2. Operations on Sets • Intersection If A and B are sets, we define their intersection as the set consisting of all elements that belong to both A and B and denoted it by A ∩ B. A ∩ B = { x | x ∈ A and x ∈ B } U A B

15. 1.2. Operations on Sets • Complement of B with respect to A If A and B are two sets, we define the complement of B with respect to A as the set of all elements that belong to A but not to B, and we denote it by A - B A - B = { x | x ∈ A and x ∉ B } U A A B

16. 1.2. Operations on Sets • Complement If U is a universal set containing A, then U-A is called the complement of A and is denoted by = {x | x ∉ A} U A

17. 1.2. Operations on Sets • Symmetric difference If A and B are two sets, we define their symmetric difference as the set of all elements that belong to A or to B, but not to both A and B, and we denote it by A B A B = {x | (x ∈A and x ∉ B) or (x ∈B and x ∉ A) } U A B

18. 1.2. Operations on Sets • Commutative Properties A U B = B U A ; A ∩ B = B ∩ A • Associative Properties A U (B U C) = ( A U B ) U C A ∩ (B ∩ C) = ( A ∩ B ) ∩ C • Distribution Properties A ∩ (B U C) = ( A ∩ B ) U ( A ∩ C ) A U (B ∩ C) = ( A U B ) ∩ ( A UC )

19. 1.2. Operations on Sets • Idempotent Properties A U A =A ; A ∩ A = A • Properties of the complement De Morgan’s Law

20. 1.2. Operations on Sets • Properties of a Universal set A U U = U A ∩ U = A • Properties of the empty set A U = A A ∩ =

21. 1.2. Operations on Sets • How to proof above properties? e.g. Proof: Proof: suppose x ∈ , then we have x ∉ A∩B, so x∈ or x ∈ , which means that x ∈ . Thus, ⊆ Conversely, suppose x ∈ , then we have x ∉ A or x ∉ B , so x ∉ A ∩ B, which means that x ∈ .Thus ⊆ Therefore, A common style of proof for statements about sets is to choose an element in one of the sets and see what we know about it.

22. 1.2. Operations on Sets • Addition Principle Theorem 2: If A and B are finite sets, then |A U B| = |A| + |B| - |A ∩ B | A ∩ B U A B

23. 1.2. Operations on Sets • Example 9 A computer company wants to hire 25 programmers to handle systems programming jobs and 40 programmers for applications programming. Of those hired, 10 will be expected to perform jobs of both types. How many programmers must be hired? (at least? ) Solution: A: the set of system programmers hired B: the set of applications programmers hired, then |A| = 25, |B| = 40, |A ∩ B| =10 |A U B| = |A| + |B| - |A ∩ B | = 25 + 40 -10 =55

24. 1.2. Operations on Sets • Generalized case for three sets Theorem 3: Let A, B and C be finite sets. Then |A U B U C| = |A| + |B| + |C| - |A∩B| - |B∩C|-|A∩C| + |A∩B∩C| A∩B A∩B∩C B A C A∩C B∩C

25. 1.2. Operations on Sets • Homework ex. 4, ex. 10, ex. 12, ex. 35 ex. 46, ex. 47

26. 1.3. Sequences • 1.4. Properties of Integers • 1.5. Matrices

27. 1.6. Mathematical Structures • Mathematical structure (system) Such a collection of objects with operations defined on them and the accompanying properties form a mathematical structure or system, for instance, Example 1: The collection of sets with the operations of union, intersection and complement and their accompanying properties is a mathematical structure. Denoted by (sets, U, ∩ , -)

28. 1.6. Mathematical Structures • Binary operation An operation that combines two objects • Unary operation An operation that requires only one object Example: the structure (5x5 matrices, +, *, T) the operations + and * are binary operations the operation T is a unary operation

29. 1.6. Mathematical Structures • Closure A structure is closed with respect to an operation if that operation always produces another/same member of the collection of objects. Example 3: The structure (5x5 matrices, +, *, T) is closed with respect to +, * and T. (why?) Example 4: The structure (odd integers, +, *) is closed with respected to *, while it is not closed with respected to +. (why?)

30. 1.6. Mathematical Structures • Commutative property If the order of the objects does not affect the outcome of a binary operation, we say that the operation is commutative , namely if x □ y = y □ x, where □ is some binary operation with commutative property. Example 6 (a) Join and meet for Boolean matrices are commutative operations A V B =B V A and A ^ B = B ^ A (b) Ordinary matrix multiplication is not a commutative operation. AB ≠ BA

31. 1.6. Mathematical Structures • Associative property if □ is a binary operation, then □ is associative or has associative property if (x □ y) □ z = x □ (y □ z) Example 7 Set union is an associative operation, since (A U B) U C = A U (B U C) is always true

32. 1.6. Mathematical Structures • Distributive property If a mathematical structure has tow binary operations, say □ and ∇, a distributive property has the following pattern: x □ (y ∇ z) = (x □ y) ∇ ( x □ z ) we say that □ distributes over ∇ Example 8 (b) the structure (sets, U, ∩, -) has two distributive properties: A U (B ∩ C) =(AUB) ∩ (AUC) A ∩ (B UC) =(A ∩ B) U (A ∩ C)

33. 1.6. Mathematical Structures • De Morgan’s law If the unary operation is ○ and the binary operation □ and ∇, then De Morgan’s law are (x □ y) ○ =x ○ ∇ y ○ , (x ∇ y) = x ○ □ y ○ Example 9 (a) Union, intersection and complement (b) The structure (real numbers, +, *, sqrt) does not satisfy De Morgan’s law (why?)

34. 1.6. Mathematical Structures • Identify If a structure with a binary operation □ contain an element e, satisfying that x □e =e□x = x for all x in the collection we call e an identify for the operation □ Example 10: For (n-by-n matrices, +,*, T), In is the identify for matrix multiplication and the n-by-n zero matrix is the identify matrix addition.

35. 1.6. Mathematical Structures • Theorem 1: If e is an identify for a binary operation □, then e is unique. Proof: Assume i is another object with identify property, then we have i □ e = e□ i = e; since e is also an identify for □, then we have i □ e =e □ i = i, therefore e = i, which means that there is at most one object with the identify property for □.

36. 1.6. Mathematical Structures • Inverse If a binary operation □ has an identity e, we say y is a □-inverse of x if x □y=y □x=e Example 11: (a) In the structure (3-by-3 matrices, +, *, T), each matrix A=[aij] has +-inverse(additive inverse), -A=[-aij]. (why ?) (b) In the structure (integers, +, *), only the integers 1 and -1 have multiplicative inverses. (why?)

37. 1.6. Mathematical Structures • Theorem 2: If □is an associative operation and x has a □-inverse y, then y is unique. Proof: Assume there is another □-inverse for x, say z, then (z □ x) □ y = e □ y = y, and z □ (x □ y) =z □ e =z since □ us associative, (z □ x) □ y = z □ (x □ y) and so y=z, which means that y is unique.

38. 1.6. Mathematical Structures • Homework ex. 4, ex. 8, ex. 13, ex. 21, ex. 32