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Chapter 1 - 2

Elementary Notions and Notations

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Section 1.3 Ordered Structures

• Tuples: Have order and can have repetitions.

• For example, (6, 7, 6) is a 3-tuple and ( ) is the empty tuple.

• We write (x1, …, xn) = (y1, …, yn) to mean xi = yi for 1 ≤ i ≤ n.

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Cartesian Product

• Cartesian Product: A ╳ B = {(x, y) | x ∊ A and y ∊ B}.

• The definition extends to more than two sets. For example, A ╳ B ╳ C = {(x, y, z) | x ∊ A, y ∊ B, z ∊ C}.

• Notation: A0 = {( )}, A1 = {(x) | x ∊ A}, and in general, An = {(x1, …, xn) | xi ∊ A}.

• Quiz (1 minute). Does (A ╳ B) ╳ C = A ╳ (B ╳ C)?

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Lists

• Lists: Are like tuples but there is no random access. • For example, <a, b, a, c> is a list with 4 elements and <>

is the empty list.• The operations on lists are head, tail, and cons. For

example, head(<a, b, a, c>) = atail(<a, b, a, c>) = <b, a, c>cons(a, <b, a, c>) = <a, b, a, c>

• The set of lists whose elements are in A is denoted by lists(A).

• Quiz (1 minute). For L = <<a>, b, <c, d>>, find head(L) and tail(L).

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Strings

• Strings: Are like lists, but are represented as juxtaposed elements from a given alphabet.

• For example, if A = {a, b}, then some strings over A are: a, b, aa, ab, ba, bb, aaa, bbb.

• The empty string is denoted by ∧. • The concatenation of two strings is their juxtaposition.

For example, the concatenation of ab and bab is abbab.

• For any string s we have s∧ = ∧s = s. • If s is a string, sn denotes the concatenation of s with

itself n times. Also s0= ∧. For example, (ab)3 = ababab.

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Languages

• A language is a set of strings, usually taken over some alphabet.

• Notation: If A is an alphabet, then the set of all strings over A is denoted by A*.

• Example. Some languages over A are: Ø, {∧}, A, and A*.

• Example. {abna | n ∊ N} = {aa, aba, abba, abbba, … } is a language over {a, b}.

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Language Operations

• Let L and M be languages. The product of L and M, denoted LM, is the language

LM = {st | s ∊ L and t ∊ M}.

• Notation: L0 = {∧}; and Ln = {s1…sn| si ∊ L}.

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Quizzes

• Quiz (1 minute). What are the products LØ and L{∧}?

• Quiz (1 minute). Solve for L in the equation {∧, a, b}L = {∧, a, b, aa, ba, aba, bba}.

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closure

• The closure L* is the set of all possible concatenations of strings in L. So

L* = L0 U L1 U … U Ln U …

• Quiz (1 minute). What are {∧}* and Ø*?

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Example

• Examine the structure of an arbitrary string x ∊ L*(ML)*.• A solution: Use the definitions to write x in terms of

strings in L and M.1. Since x ∊ L*(ML)*, it follows that x = uv where u ∊ L* and

v ∊ (ML)*.

2. Since u ∊ L*, either u = ∧ or u = s1…sn for some n where si ∊ L.

3. Since v ∊ (ML)*, either v = ∧ or v = r1t1 …rktk for some k where ri ∊ M and ti ∊ L.

So x has one of four forms: ∧, s1…sn, r1t1 …rktk, or s1…snr1t1 …rktk .

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Relations

• A relation is a set of tuples. If R is a relation and (x1, …, xn) ∊ R, we write R(x1, …, xn).

• We can usually represent a relation as a subset of some cartesian product.

• Example. Let R = {(0, 0), (1, 1), (4, 2), (9, 3), …, (n2, n), …} = {(n2, n) | n ∊ N}. We might call R the “is square of” relation on N. Notice also that R ⊂ N ╳ N.

• Notation: If R is binary, we can use infix to represent pairs in R. For example, from the previous example, we have (9, 3) ∊ R. So we can also write R(9, 3) or 9 R 3 or 9 is square of 3.

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Relational Databases

• A relational database is a relation where the indexes of a tuple have associated names called attributes.

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Example

• Let Students = {(x, y, z) | x is a Name, y is a Major, and z is Credits}.

• Who are the people majoring in CS? {x | (x, CS, z) ∊ Students, for some z}. Note: We need “for some z” to indicate that z is a

variable. • How many math majors are upper division students?

| {x | (x, math, z) ∊ Students and z ≥ 90} |. • What is the major of AbeLincoln?

{y | (AbeLincoln, y, z) ∊ Students, for some z}. • What is the history department database of names and

their credits? {(x, z) | (x, history, z) ∊ Students}.

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Counting Tuples (or strings or lists)

• Product Rule: | A ╳ B | = | A | | B | and | An | = | A |n.

• Example. If A = {a, b} and B = {1, 2, 3}, then

A ╳ B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}.

So | A ╳ B | = 6 = (2)(3) = | A | | B |.

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Example• Count the number of strings of

length 8 over A = {a, b, c} that begin with either a or c and have at least one b.

• A Solution: Split the problem up into easier problems and combine the results (divide and conquer).

• Let U be the universe consisting of the strings over A of length 8 that begin with either a or c.

• Let B be the subset of U consisting of strings with no b’s.

• Then the set of strings to count is U – B, as pictured.

U-B U

B

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Example (cont’d)

• It is easy to calculate the cardinality of U – B: | U – B | = | U | – | U ∩ B | = | U | – | B | (since B is a subset of U)

• It is also easy to count U because it has the same cardinality as the set {a, c} ╳ A7, which is | {a, c} ╳ A7 | = | {a, c} | | A7 | = | {a, c} | | A |7 = (2)37.

• It is also easy to count B because it has the same cardinality as the set {a, c}8, which is | {a, c}8 | = | {a, c} |8 = 28.

• So we have the answer: | U – B | = | U | – | U ∩ B | = | U | – | B | = (2)37 – 28, which is 4118.

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Section 1.4 Graphs and Trees

• A graph is set of objects called vertices or nodes where some pairs of objects may be connected by edges. (A directed graph has edges that point in one direction.)

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Example

• Draw a graph of the South American countries that touch the Pacific Ocean and their neighbors, where the vertices are countries and an edge indicates a common border.

• Vertices = {Co, V, E, Br, P, Bo, Ch, A}

• Edges = {{Co, V}, {Co, E}, ….}.

CoV

E

P

Ch

Bo

Br

A

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Some definitions

• A path from vertex x0 to xn is a sequence of edges that we denote by vertices x0, x1, …, xn, where there is an edge from xi–1 to xi for 1≤ i ≤ n.

• The length of a path is the number of edges. • A cycle is a path with distinct edges that begins

and ends at the same vertex. • Example. A, Bo, A, is not a cycle since the

edge{A, Bo} occurs twice. A, Bo, Br, A, is a cycle.

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Quiz (1 minute)

• What is a longest path from A to V with distinct edges and no cycles?

• Answer: The length is 6. For example, A, Bo, Br, P, E, Co, V.

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n-colorable

• A graph is n-colorable if it’s vertices can be colored with n colors with distinct colors for adjacent vertices. The chromatic number of a graph is the smallest such n.

• Quiz (1 minute). What is the chromatic number of the example graph?

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Graph Traversals

• A graph traversal starts at some vertex v and visits all vertices x that can be reached by a path from v to x. But don’t visit any vertex more than once.

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Breadth-First

• If the graph has n vertices then start with a vertex v and do the following:

for k := 0 to n – 1 do visit(v, k) od

where visit(v, k) visits all x not visited if there is a path from v to x of length k.

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Quizzes

• Use the pictured graph for the following quizzes.

• Quiz (1 minute). Find a breadth-first traversal that starts at F.

• One answer: F, H, D, G, B, A, E, C.• Quiz (1 minute). Find a breadth-

first traversal that starts at C.• One answer: C, A, E, D, B, F, H, G.

A B

D

F

HG

E

C

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Depth-First

• Start at a vertex v and call the procedure D(v), which is defined as follows:

D(v): if v has not been visited then

visit(v);

for each edge from v to x

do D(x) od

fi

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Quizzes

• Quiz (1 minute). Find a depth-first traversal of the pictured graph that starts at F.

• One answer: F, H, G, D, B, A, C, E.

• Quiz (1 minute). Find a depth-first traversal of the pictured graph that starts at E.

• One answer: E, D, F, H, G, A, C, B.

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Trees• A tree is a connected graph (a path

between any two points) with no cycles. Most trees are oriented so that they look like upside-down trees, such as the tree pictured.

• The top node is the root, the nodes directly below a node are its children, the node directly above a node is the parent, the bottom nodes are leaves, and the height or depth of the tree is the length of the longest path of distinct edges from root to a leaf.

• Example. For this tree the root is A. The children of A are B, C, D. D is the parent of G. The height or depth of the tree is 3. The leaves are E, F, C, H, I.

A

B C D

E F G

H I

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A Recursive Definition

• Any node of a tree is the root of a subtree. • One way to represent a tree is as a list whose

head is the root of the tree and whose tail is the list of subtrees, where each subtree is represented in the same way.

• Example. The pictured tree can be represented by the list

<A, <B, <E>, <F>>, <C>, <D, <G, <H>, <I>>>>.

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Expression Tree

• Any algebraic expression can be represented as a tree.

• For example, the tree for the expression (x – y) + log(z + w) is pictured to the right.

• Quiz (1 minute). Do a depth-first (left to right) traversal.

• Answer: + – x y log + z w. This is the prefix form of the expression.

+

- log

x y +

z w

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Binary Trees

• A binary tree is either empty, denoted by <>, or each node has two subtrees that are binary trees and are called the left and right subtrees of the node.

• If a binary tree is not empty, we’ll represent it as a list of the form <L, x, R>, where x is the root and L and R are the left and right subtrees, respectively.

• Example. The binary tree with a single node x is denoted by <<>, x, <>>.

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binary search tree

• A binary search tree represents ordered information, where the predecessors and successors of a node are in its left and right subtrees, respectively.

• Example. A binary search tree for the first six prime numbers is pictured.

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3 11

2 5 13

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Spanning Trees

• A spanning tree for a connected graph is a tree whose nodes are the nodes of the graph and whose edges are a subset of the edges of the graph.

• A minimal spanning tree minimizes the sum of weights on the edges of all spanning trees.

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Example

• Use Prim’s algorithm to construct a minimal spanning tree for the pictured graph, starting with node D.

• Solution: A minimal spanning tree is constructed in 4 steps

A B

C

D

E

3

1 32

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1

B

D

1 1 1 1

B B B

D D DC C C

2 22

A A

11

E2

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The End of Chapter 1 – 2