Power Set of a set and Relations

Size: px

Start display at page:

Download "Power Set of a set and Relations"

Melvyn Caldwell
5 years ago
Views:

1 Power Set of a set and Relations 1

2 Power Set (1) Definition: The power set of a set S, denoted P(S), is the set of all subsets of S. Examples Let A={a,b,c}, P(A)={,{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c}} Let A={{a,b},c}, P(A)={,{{a,b}},{c},{{a,b},c}} Note: the empty set and the set itself are always elements of the power set.

3 Power Set (2) The power set is a fundamental combinatorial object useful when considering all possible combinations of elements of a set Fact: Let S be a set such that S =n, then P(S) = 2 n

4 Proof of P(A) =2^ A Proof: ( By induction) Base case holds: A =0, then A={} and P(A)={ }. Induction Hypothesis: Assume to be true for all sets of cardinality n-1, i.e. A =n-1 implies P(A) =2^{n-1}. For A =n Let A={a_1,a_2,,a_n}. Any subset not containing a_n is a subset of B=A\{a_n}. How many subsets of A are there not containing a_n? This is nothing but the set of all subsets of B. The number is 2^{n-1} by induction hypothesis. 4

5 Proof of P(A) =2^ A Contd How to get a subset of A containing a_n? Take a subset S of B. Let S =S {a_n}. Now S is a subset of A containing a_n. Can any subset of A containing a_n be obtained in this way? Answer is Yes!! Let S be any subset of A containing a_n. Let S=S \{a_n}. Now S is a subset of A not containing a_n. 5

6 Proof of P(A) =2^ A Contd Let X_1 be the set of all subsets of A not containing a_n Let X_2 be the set of all subsets of A containg a_n. Now P(A)= X_1 X_2 and X_1 X_2 =. Also S S =S {a_n} is a bijection. This implies S = S =2^{n-1} by induction hypothesis So P(A)) = S + S = 2^{n-1} +2^{n-1}=2^{n}. 6

7 How to generate all subsets of a Set of n elements? 7

8 Computer Representation of Sets (1) There really aren t ways to represent infinite sets by a computer since a computer has a finite amount of memory If we assume that the universal set U is finite, then we can easily and effectively represent sets by bit vectors Specifically, we force an ordering on the objects, say: U={a 1, a 2,,a n } For a set A U, a bit vector can be defined as, for i=1,2,,n b i =0 if a i A b i =1 if a i A

9 Computer Representation of Sets (2) Examples Let U={0,1,2,3,4,5,6,7} and A={0,1,6,7} The bit vector representing A is: How is the empty set represented? How is U represented? Set operations become trivial when sets are represented by bit vectors Union is obtained by making the bit-wise OR Intersection is obtained by making the bit-wise AND

10 BITWISE OPERATOR (IN C) Operators & Meaning of operators Bitwise AND Bitwise OR ^ Bitwise exclusive OR ~ Bitwise complement << Shift left >> Shift right

11 Computer Representation of Sets (3) Let U={0,1,2,3,4,5,6,7}, A={0,1,6,7}, B={0,4,5} What is the bit-vector representation of B? Compute, bit-wise, the bit-vector representation of A B Compute, bit-wise, the bit-vector representation of A B What sets do these bit vectors represent?

12 Computer Representation of Sets Array representations: U={1,2,.n} S can be represented by an array A such that A[i]=1 if i S else A[i]=0. Disjoint Set operations: Union, j=find(i) if I in S_j. 12

13 Programming Question Using bit vector, we can represent sets of cardinality equal to the size of the vector What if we want to represent an arbitrary sized set in a computer (i.e., that we do not know a priori the size of the set)? What data structure could we use? Tree representation of disjoint Set: To be discussed after Graphs are introduced.

14 Relations 14

15 Relations and Their Properties To represent relationships between elements of sets Ordered pairs: binary relations Definition 1: Let A and B be sets. A binary relation from A to B is a subset of A B. a R b: (a,b) R a is related to b by R R a b: (a,b) R Ex.1-3 Functions as relations Relations are generalization of functions

16 FIGURE 1 (8.1) FIGURE 1 Displaying the Ordered Pairs in the Relation R from Example 3.

17 Relations on a Set Definition 2: a relation on the set A is a relation from A to A. How many relations are there on a set with n elements? ( Such questions will be discussed in Counting and advanced counting)

18 FIGURE 2 Displaying the Ordered Pairs in the Relation R from Example 4.

19 Properties of Relations Definition 3: A relation R on a set A is called reflexive if (a,a) R for every element a A. a((a,a) R) Definition 4: A relation R on a set A is called symmetric if (b,a) R whenever (a,b) R, for all a,b A. A relation R on a set A such that for all a,b A, if (a,b) R and (b,a) R, then a=b is called antisymmetric. a b((a,b) R (b,a) R) a b(((a,b) R (b,a) R) (a=b)) Not opposites

20 Definition 5: A relation R on a set A is called transitive if whenever (a,b) R and (b,c) R, then (a,c) R, for all a,b,c A. a b c(((a,b) R (b,c) R) (a,c) R)

21 Combining Relations Two relations can be combined in any way two sets can be combined Definition 6: R: relation from A to B, S: relation from B to C. The composite of R and S is the relation consisting of ordered pairs (a,c), where a A, c C, and there exists an element b B such that (a,b) R and (b,c) S. (denoted by S R)

22 Definition 7: Let R be a relation on the set A. The powers R n, n=1,2,3,, are defined recursively by R 1 =R, and R n+1 =R n R. Theorem 1: The relation R on a set A is transitive iff R n R for n=1,2,3, Proof.

23 Representing Relations To represent binary relations Ordered pairs Tables Zero-one matrices Directed graphs

24 Representing Relations Using Matrices Suppose that R is a relation from A={a 1,a 2,,a m } to B={b 1,b 2,,b n }. R can be represented by the matrix M R =[m ij ], where m ij =1 if (a i,b j ) R, or 0 if (a i,b j ) R.

25 FIGURE 1 The Zero-One Matrix for a Reflexive Relation.

26 FIGURE 2 (8.3) FIGURE 2 The Zero-One Matrices for Symmetric and Antisymmetric Relations.

27 Representing Relations Using Digraphs Definition 1: A directed graph (digraph) consists of a set V of vertices (or nodes) and a set E of ordered pairs of elements of V called edges (or arcs). The vertex a is called the initial vertex of the edge (a,b), and the vertex b is called the terminal vertex of this edge.

28 FIGURE 3 The Directed Graph.

29 FIGURE 4 (8.3) FIGURE 4 The Directed Graph of the Relations R.

30 FIGURE 5 (8.3) FIGURE 5 The Directed Graph of the Relations R.

31 FIGURE 6 (8.3) FIGURE 6 The Directed Graph of the Relations R and S.

32 Closures of Relations Ex. Telephone line connections Let R be a relation on a set A. R may or may not have some property P. If there is a relation S with property P containing R such that S is a subset of every relation with property P containing R, then S is called the closure of R with respect to P. S: the smallest relation that contains R

33 Closures Reflexive closure Ex: R={(1,1),(1,2),(2,1),(3,2)} on the set A={1,2,3} The smallest relation contains R? The reflexive closure of R=R, where ={(a,a) a A} Symmetric closure The symmetric closure of R=R R -1, where R -1 ={(b,a) (a,b) R} Transitive closure

34 Paths in Directed Graphs Definition 1: A path from a to b in the directed graph G is a sequence of edges (x0,x1), (x1,x2),,(xn-1,xn) in G, where n is a nonnegative integer, and x0=a and xn = b. The path denoted by x0,x1,,xn-1, xn has length n. A path of length n >=1 that begins and ends at the same vertex is called a circuit or cycle. There is a path from a to b in R if there is a sequence of elements a, x1, x2,, xn-1, b with (a,x1) R, (x1,x2) R,, and (xn-1,b) R.

35 FIGURE 1 A Directed Graph.

36 Theorem 1: Let R be a relation on a set A. There is a path of length n, where n is a positive integer, from a to b iff (a,b) R n. Proof. (by mathematical induction)

37 Transitive Closures Definition 2: Let R be a relation on a set A. The connectivity relation R* consists of pairs (a,b) such that there is a path of length at least one from a to b in R. R*= n=1.. R n.

38 Theorem 2: The transitive closure of a relation R equals the connectivity relation R*. Proof. Lemma 1: Let A be a set with n elements, and R be a relation on A. If there is a path of length at least one in R from a to b, then there is such as path with length not exceeding n. When a b, if there is a path of length at least one in R from a to b, then there is such as path with length not exceeding n-1. Proof.

39 FIGURE 2 Producing a Path with Length Not Exceeding n.

40 Theorem 3: Let M R be the zero-one matrix of the relation R on a set with n elements. Then the zero-one matrix of the transitive closure R* is M R* = M R M R [2] M R [3] M R [n].

41 Algorithm 1: A procedure for computing the transitive closure. Procedure transitive closure (M R : zero-one n*n matrix) A:= M R B:=A for i:=2 to n begin A:=A M R B:=B A end O(n 4 ) bit operations

42 Warshall s Algorithm Named after Stephen Warshall in n 3 bit operation Also called Roy-Warshall algorithm, Bernard Roy in 1959 Interior vertices W0, W1,, Wn Wk=[w ij (k) ], where w ij (k) =1 if there is a path from v i to v j such that all interior vertices of this path are in the set [v1, v2,, vk]. Wn= M R*

43 FIGURE 3 The Directed Graph of the Relations R.

44 Observation: we can compute Wk directly from Wk-1 Two cases (Fig. 4) There is a path from vi to vj with its interior vertices among the first k-1 vertices w ij (k-1) =1 There are paths from vi to vk and from vk to vj that have interior vertices only among the first k-1 vertices w ik (k-1) =1 and w kj (k-1) =1

45 FIGURE 4 Adding v k to the Set of Allowable Interior Vertices.

46 Lemma 2: Let Wk=w ij (k) be the zero-one matrix that has a 1 in its (i,j)th position iff there is a path from vi to vj with interior vertices from the set {v1, v2,, vk}. Then w ij (k) =w ij (k-1) (w ik (k-1) w kj (k-1) ), whenever I, j, and k are positive integers not exceeding n.

47 Algorithm 2: Warshall Algorithm Procedure Warshall(M R : n*n zero-one matrix) W:= M R for k:=1 to n begin for i:=1 to n begin for j:=1 to n wij:=wij (wik wkj) end end

48 Equivalence Relations Definition 1: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. Definition 2: Two elements a and b are related by an equivalence relation are called equivalent. The notation a~b: to denote that a and b are equivalent elements with respect to a particular equivalence relation

49 Equivalence Classes Definition 3: Let R be an equivalence relation on a set A. The set of all elements that are related to an element a of A is called the equivalence class of a. Denoted by [a] R or [a] [a] R = {s (a,s) R} If b [a] R, then b is called a representative of the equivalence class

50 Equivalence Classes and Partitions Theorem 1: Let R be an equivalence relation on a set A. These statements for elements a and b of A are equivalent: a R b [a]=[b] [a] [b] Proof.

51 Observations a A [a] R =A [a] [b]= when [a] R [b] R The equivalence classes form a partition of A A partition of a set S is a collection of disjoint nonempty subsets of S that have S as their union Ai for i I, Ai Aj= when i j i I Ai=S

52 FIGURE 1 A Partition of a Set.

53 Theorem 2: Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S. Conversely, given a partition {Ai i I} of the set S, there is an equivalence relation R that has the sets Ai, i I, as its equivalence classes.

54 Partial Orderings Definition 1: A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive. A set S together with its partial ordering R is called a partially ordered set, or poset, (S, R). Proof. Ex.1-3:,, Ex.4: older than Notation: : a b, but a b

55 Definition 2: The elements a and b of a poset (S, ) are called comparable if either a b or b a. If neither a b nor b a, a and b are incomparable. ( partial ) Ex. 5 When every two elements in the set are comparable, the relation is called a total ordering

56 Definition 3: If (S, ) is a poset and every two elements of S are comparable, S is called a totally ordered or linearly ordered set (or a chain), and is called a total order or a linear order. Ex.6: (Z, ) Ex.7: (Z+, ) Definition 4: (S, ) is a well-ordered set if it is a poset such that is a total ordering and every nonempty subset of S has a least element. Ex.8

57 Theorem 1: (The Principle of Well-Ordered Induction) Suppose that S is a well-ordered set. Then P(x) is true for all x S, if (inductive step): For every y S, if P(x) is true for all x S with x y, then P(y) is true. Proof.

58 Lexicographic Order (a 1, a 2 ) (b 1, b 2 ) (A 1, 1), (A 2, 2) Either if a 1 1 b 1 Or if a 1 =b 1 and a 2 2 b 2 n-tuples (Fig. 1)

59 FIGURE 1 The Ordered Pairs Less Than (3,4) in Lexicographic Order.

60 Hasse Diagrams Hasse Diagrams: some edges do not have to be shown Reflexive: loops can be removed Transitive: if (a,b) and (b,c), then (a,c) can be removed Remove all the arrows

61 FIGURE 2 Constructing the Hasse Diagram for ({1,2,3,4}, ).

62 FIGURE 3 (8.6) FIGURE 3 Constructing the Hasse Diagram of ({1,2,3,4,6,8,12}, ).

63 FIGURE 4 The Hasse Diagram of (P({a,b,c}), ).

64 Maximal and Minimal Elements a is maximal in the poset (S, ) if there is no b S such that a b. a is minimal in the poset (S, ) if there is no b S such that b a.

65 FIGURE 5 The Hasse Diagram of a Poset.

66 FIGURE 6 Hasse Diagrams of Four Posets. FIGURE 6 (8.6)

67 In a subset A of a poset (S, ), if u is an element of S such that a u for all a A, then u is an upper bound of A. In a subset A of a poset (S, ), if l is an element of S such that l a for all a A, then l is an lower bound of A.

68 FIGURE 7 The Hasse Diagram of a Poset.

69 Lattices A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called a lattice

70 FIGURE 8 Hasse Diagrams of Three Posets. FIGURE 8 (8.6)

71 Topological Sorting Topological sorting: constructing a compatible total ordering from a partial ordering A total ordering is said to be compatible with the partial ordering R if a b whenever a R b. Lemma 1: Every finite nonempty poset (S, ) has at lease one minimal element.

72 Algorithm 1: topological sorting Procedure topological sort(s, : finite poset) k:=1 while S begin ak:=a minimal element of S S:=S-{ak} k:=k+1 end

73 FIGURE 9 (8.6) FIGURE 9 A Topological Sort of ({1,2,4,5,12,20}, ).

74 FIGURE 10 The Hasse Diagram for Seven Tasks.

75 FIGURE 11 A Topological Sort of the Tasks. FIGURE 11 (8.6)

CS 441 Discrete Mathematics for CS Lecture 24. Relations IV. CS 441 Discrete mathematics for CS. Equivalence relation

CS 441 Discrete Mathematics for CS Lecture 24 Relations IV Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Equivalence relation Definition: A relation R on a set A is called an equivalence relation