Mathematical Preliminaries

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CSCI 2670

CSCI 2670 Mathematical Preliminaries Outline for Today

Material from Chapter 0.

I Sets

I Functions

I Relations

I Graphs

CSCI 2670 Mathematical Preliminaries Sets

I A set is an unordered collection of objects. The objects in the set are called its elements or members.

I In writing, uppercase Roman letters and braces {...} indicate sets. S = {1, 2, 3, 5, 8, 13, 21}

I a ∈ S (a 6∈ S) indicates a is (is not) a member of S.

I For sets too big to list, we use “...” or state a rule.

I S = {2, 4, 6, 8 ...} I T = {x|x is a positive, even integer}

I Order and repetition do not matter with sets.

I {1, 3, 5, 7, 9} and {3, 1, 7, 9, 5} denote the same set. I {a, a, b, b, c, c} and {a, b, c} denote the same set.

CSCI 2670 Mathematical Preliminaries Common Sets of Numbers

I N = {0, 1, 2,...}, the natural numbers.

I Z = {0, ±1, ±2,...}, the integers.

p I Q = { q | p ∈ Z and q ∈ Z, q 6= 0}, the rational numbers.

I R, the real numbers (numbers with inﬁnite decimal representations).

I Some authors use N, Z, Q, R. + I Some use superscripts: Z = {1, 2,...} (the positive integers).

I Some don’t consider 0 a natural number.

CSCI 2670 Mathematical Preliminaries Subsets and Supersets

I If all members of A are members of B, then A is a subset of B.

I A ⊆ B: A is a subset of B. I A * B: A is not a subset of B.

I If A ⊆ B, then B is a superset of A.

I If A ⊆ B and B ⊆ A, then A = B.

I If A ⊂ B and A 6= B, then A is a proper subset of B.

I Observe that for any set S, S ⊆ S and S 6⊂ S.

CSCI 2670 Mathematical Preliminaries Subsets and Supersets

I The empty set ∅ (or ∅, or {}) is the set with no members. I For any set S, ∅ ⊆ S.

I Note: {∅}= 6 ∅.

I List all of the proper subsets of the following:

I {a}

I {a, b}

I {a, b, c}

I {{a, {b}, {c}}

I ∅

CSCI 2670 Mathematical Preliminaries Cardinality

I The cardinality |S| of set S is the number of elements in S.

I For ﬁnite sets, |S| will be an integer.

S = ∅ |S| = 0 S = {a, a, a} |S| = 1 S = {a, b, c} |S| = 3 S = {{a}, {b, c}} |S| = 2 S = {x| x ∈ N} |S| is “countably inﬁnite”

CSCI 2670 Mathematical Preliminaries Powersets

I The powerset P(S) of set S is the set of all subsets of S.

n I In general, if |S| = n, then |P(S)| = 2 .

S |S| P(S) |P(S)| ∅ 0 ? ? {a} 1 ? ? {a, b} 2 ? ? {a, b, c} 3 ? ? {a, b, c, d} 4 ? ?

CSCI 2670 Mathematical Preliminaries Powersets

I The powerset P(S) of set S is the set of all subsets of S.

n I In general, if |S| = n, then |P(S)| = 2 . S |S| P(S) |P(S)| ∅ 0 {∅} 1 {a} 1 {∅, {a}} 2 {a, b} 2 {∅, {a}, {b}, {a, b}} 4 {a, b, c} 3 {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} 8 {a, b, c, d} 4 {∅, {a}, {b}, {c}, {d}{a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, 16 {b, c, d}, {a, b, c, d}}

CSCI 2670 Mathematical Preliminaries ordered n-tuples

I Let a1, a2, ... an be elements of a set.

I An ordered list of these elements is a sequence.

I (a1, a2,..., an) is called an ordered n-tuple.

I (ai , aj ) is an ordered pair.

I The order of the elements matters.

I (ai , aj ) = (aj , ai ) if and only if ai = aj . I If ai 6= aj , then (ai , aj ) 6= (aj , ai ).

I In a sequence or n-tuple, repetitions can occur: (1,1,2,3,5,8).

CSCI 2670 Mathematical Preliminaries Cartesian Products

I A × B is the Cartesian product (cross product) of sets A and B.

A × B = {(a, b)| a ∈ A and b ∈ B}

I N-ary Cartesian product: If A1, ..., An are sets,

A1 × ... × An = {(a1,..., an)| ai ∈ Ai for each i = 1,..., n}

n I A is shorthand for A × A × ... × A.

I Note that × is not commutative or associative.

I A × B 6= B × A. I A × (B × C) 6= (A × B) × C.

CSCI 2670 Mathematical Preliminaries Cartesian Products

I Let A = {a, b, c} and B = {0, 1}

I What is A × A, A × B, B × A, and B × B.

I Let R be the set {(1, 1)}. Is R a subset of any of the above?

I Let R be the set {(b, 1), (1, 1)}. Is R a subset of any of the above?

CSCI 2670 Mathematical Preliminaries Venn Diagrams

I Sets can be represented using Venn diagrams.

I A rectangle represents U, the universal set (the set of everything).

I Sets are drawn using closed shapes inside of U.

I Points inside each shape represent elements of the set.

I (We may cross out a region to indicate that it is empty).

I bob ∈ A, A ⊂ U.

CSCI 2670 Mathematical Preliminaries Venn Diagrams

I Draw a Venn diagram showing that A ⊆ B. I Draw a Venn diagram showing that A * B. I Draw a Venn diagram showing that A * B and B * A. I Draw a Venn diagram showing that A ⊆ B and A ⊆ C but B 6= C.

CSCI 2670 Mathematical Preliminaries Set Operations

I Union: A ∪ B = {x|x ∈ A OR x ∈ B}

I Intersection: A ∩ B = {x|x ∈ A AND x ∈ B} I If A ∩ B = ∅, then A and B are disjoint.

I Set Diﬀerence: A − B = {x|x ∈ A AND x ∈/ B}

I Complement: A = U − A

CSCI 2670 Mathematical Preliminaries Generalized Union and Intersection

Let A1, A2, ..., An be sets. n [ I Ai = A1 ∪ A2 ∪ ... ∪ An i=1

n \ I Ai = A1 ∩ A2 ∩ ... ∩ An. i=1

Example: Let A1 = {1, 2, 3}, A2 = {3, 4, 5}, A3 = {1, 3, 6}. 3 [ I Ai = {1, 2, 3, 4, 5, 6} i=0 3 \ I Ai = {3} i=0

CSCI 2670 Mathematical Preliminaries Questions

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4, 5}, B = {0, 3, 6}. What is:

I A ∪ B

I A ∩ B

I A − B

I B − A

I A

I U

I A ∩ U

CSCI 2670 Mathematical Preliminaries Functions

I Let A and B be nonempty sets.

I A function f from A to B (f : A → B) is an assignment of exactly one element in B to each element in A.

I A is the domain of f . I B is the codomain of f . I {f (a)|a ∈ A} is the range of f .

I f is sometimes called a map, mapping, or transformation.

I If a ∈ A, f (a) denotes the element in B assigned to a.

I If f : A → B is deﬁned for each a ∈ A, then f is a total function.

I If there is an a ∈ A with f (a) undeﬁned, f is a partial function.

I Functions can be written in preﬁx notation—g(x, y)—or inﬁx notation—x + y.

CSCI 2670 Mathematical Preliminaries Functions

I Given function f : A → B:

I A = {a, b, c, d}.

I B = {0, 1, 2, 3}.

a f (x) a 1 b 0 c 0 d 1

I What’s the domain of f ?

I What’s the codomain of f ?

I What’s the range of f ?

I How would you represent f as a set?

CSCI 2670 Mathematical Preliminaries Injective, Surjective, and Bijective Functions

Let f : A → B be a function. f is I injective (one-to-one) if and only if each element of A gets mapped to a unique element of B.

I surjective (onto) if and only if the codomain of f equals the range of f .

I bijective (a one-to-one correspondence) if it is injective and surjective. Important! if f : A → B is bijective, then |A| = |B|. Is this function bijective? Assume A = {a, b, c, d} and B = {0, 1, 2, 3}. a f (x) a 1 b 0 c 0 d 1

CSCI 2670 Mathematical Preliminaries Which are Bijective?

I Assume f : R → R.

I f (x) = 3x

I f (x) = sin(x) + 20 2 I f (x) = x + 1 3 I f (x) = x

+ + I What if f : R → R ?

I Suppose g : N → R. Can g be bijective?

CSCI 2670 Mathematical Preliminaries Unary, Binary, k-ary Functions

I f : A → B is a unary function (it has one input).

I f : A1 × A2 → B is a binary function (it has two inputs). I It maps ordered pairs from A1 × A2 to elements of B.

I f : A1 × A2 × ... × Ak → B is a k-ary (or arity k) function (it has k inputs).

I It maps ordered k-tuples from A1 × A2 × ... × Ak to elements of B.

I Finite binary functions can be represented using a table.

I Let A = {1, 2, 3} and B = {2, 5}. Deﬁne f : A × B → N as: f 2 5 1 2 5 2 4 10 3 6 15

CSCI 2670 Mathematical Preliminaries Relations

I A subset of A1 × A2 × ... × Ak is a k-ary relation.

I Alternatively, a relation can be viewed as a function:

A1 × A2 × ... × Ak → {TRUE, FALSE}.

I A function whose codomain is {TRUE, FALSE} is sometimes called a predicate (sometimes a property).

I E.g., ≤ is a binary relation on R × R .

I As a set, (4, 4) ∈≤.

I As a function, ≤ (4, 4) = TRUE, but ≤ (5, 4) = FALSE.

CSCI 2670 Mathematical Preliminaries Equivalence Relations

I Let A be a set.

I An equivalence relation R on A is a relation R ⊆ A × A that is

I reﬂexive: For each x ∈ A,(x, x) ∈ R.

I symmetric: For each x, y ∈ A, if (x, y) ∈ R, then (y, x) ∈ R.

I transitive: For each x, y, z ∈ A, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.

I “=” is obviously an equivalence relation.

I So is “has the same mother as”.

CSCI 2670 Mathematical Preliminaries Equivalence Relations

I An equivalence relation R on A is a relation R ⊆ A × A that is

I reﬂexive: For each x ∈ A,(x, x) ∈ R.

I symmetric: For each x, y ∈ A, if (x, y) ∈ R, then (y, x) ∈ R.

I transitive: For each x, y, z ∈ A, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.

I Consider the following relation: x ≡7 y if and only if x − y is a multiple of 7.

I Is this an equivalence relation?

CSCI 2670 Mathematical Preliminaries Outline for Today

Material from Chapter 0.

I Graphs

I Recursive Deﬁnitions

I Strings and languages

I Logic

I Proofs For Next Week: Read Ch 1 (Sec 1.1-1.2)

CSCI 2670 Mathematical Preliminaries Graphs

I A graph G = (V , E) consists of I a nonempty set V of vertices (or nodes) and

I a set E of edges (represented as pairs of vertices from V).

I In an undirected graph, the pairs are unordered—e.g., {u, v}. In a directed graph, the pairs are ordered—e.g., (v, u).

I A loop is an edge that connects one vertex to itself.

I A graph is simple if it there are no duplicate edges between nodes.

I Let G = ({v1, v2, v3, v4}, {{v1, v2}, {v1, v3}, {v1, v4}, {v3, v4}})

CSCI 2670 Mathematical Preliminaries Subgraphs

I If G1 = (V1, E1) and G2 = (V2, E2) are graphs such that V2 ⊆ V1 and E2 ⊆ E1, then G2 is a subgraph of G1.

I H = ({v1, v3, v4}, {{v1, v3}, {v1, v4}, {v3, v4}}) is a subgraph of G.

CSCI 2670 Mathematical Preliminaries Degree of Vertices

I In an undirected graph, the degree deg(v) of a vertex v is the number of edges incident with it (loops count twice).

I Above, the deg(v2) = 3, and deg(v3) = 2.

CSCI 2670 Mathematical Preliminaries Degree of Vertices

I In a directed graph, edges are ordered and indicated using arrows.

− I The indegree deg (v) of v is the number of incoming edges.

+ I The outdegree deg (v) of v is the number of outgoing edges.

− + I Above, the deg (a) = 0 but deg (a) = 2.

CSCI 2670 Mathematical Preliminaries Paths in Undirected Graphs

I A path from u to v in an undirected, simple graph is a sequence of vertices v1, v2,..., vn such that u = v1 and v = vn, and for each vi < vn, {vi , vi+1} is an edge. A circuit (cycle) is a path in which u = v.

I A path is simple if no vertex is repeated.

I A circuit is simple if n > 2, v1 = vn, but no other vertex is repeated.

I A graph is connected if there is a path connecting every two vertices.

I The above graph is connected.

CSCI 2670 Mathematical Preliminaries Trees

I A connected graph without simple cycles is called a tree.

I Each pair of distinct nodes is connected by exactly one simple path.

I A speciﬁc node might be designated the root.

I The nodes of degree 1 (other than the root) are leaves.

CSCI 2670 Mathematical Preliminaries Paths in Directed Graphs

I A path from u to v in a directed graph G = (V , E) is a sequence of vertices v1, v2,..., vn such that u = v1 and v = vn, and for each vi < vn, (vi , vi+1) ∈ E.

I G is strongly connected if a directed path connects any two distinct vertices.

I If the underlying undirected graph is connected, G is weakly connected.

I The above graph is not strongly connected.

CSCI 2670 Mathematical Preliminaries Directed Graphs

Draw the following directed graph G = (V , E), where 1. V = {1, 2, 3, 4, 5, 6}, and 2. E = {(1, 2), (1, 5), (2, 1), (2, 4), (5, 4), (5, 6), (6, 1), (6, 3)} Is it strongly connected?

CSCI 2670 Mathematical Preliminaries Recursive Deﬁnitions

I An inﬁnite set S can be deﬁned recursively.

I Basis: A finite number of items is first specified to be in S.

I Recursive Step: A set of operations generates additional members of S from previously deﬁned members of S.

I A closure condition ensures only objects deﬁned in the basis or via ﬁnite number of applications of the operations are members of S.

CSCI 2670 Mathematical Preliminaries Recursive Deﬁnitions

I A recursive deﬁnition of N:

I Basis: 0 ∈ N.

I Recursive Step: If n ∈ N, then s(n) ∈ N.

I Only 0 and objects generated by a ﬁnite number of applications of s are in N.

I A recursively deﬁned set S can be associated with a sequence of sets S0, S1, S2 .... I S0: the basis of S.

I Si+1: Si together with items generated by a single application of operations in the recursive step.

CSCI 2670 Mathematical Preliminaries Strings and Languages

I An alphabet is a nonempty, ﬁnite set of objects (symbols).

I Σ and Γ are usually used to indicate alphabets.

I For instance, Σ1 = {a, b, c, d, e, f },Σ2 = {0, 1}

I A string over Σ is any ﬁnite sequence of symbols from Σ.

I 0, 101, and 0101111 are strings over Σ = {0, 1}.

I The length |w| of a string w is the number of symbols in w.

I |0| = 1, |101| = 3, and |0101111| = 7.

I We refer to the symbols of w by their index: w = w1w2w3 ....

I The empty string ε (or λ) consists of no symbols; |ε| = 0.

R I w = wn ... w2w1 is the reverse of w = w1w2 ... wn.

I A consecutive sequence of symbols of w is a substring of w.

CSCI 2670 Mathematical Preliminaries Strings and Languages

I Let x = x1 ... xn and y1 ... ym be strings over some alphabet.

I xy = x1 ... xny1 ... ym is the concatenation of x and y.

I Σ = {0, 1},

I x = 01, y = 001.

I xy = 01001, yx = 00101.

k I If w is a string and k ∈ N, then w is the concatenation of k w’s.

3 I If w = 001, then w = 001001001.

0 I Note: w = ε for any string w.

CSCI 2670 Mathematical Preliminaries Σ∗ and Languages

I Let Σ be some alphabet.

∗ I Σ is the set of strings deﬁned as follows: ∗ I Basis: ε ∈ Σ . ∗ ∗ I If w ∈ Σ and a ∈ Σ, then wa ∈ Σ .

I Nothing not in the basis or else formed by a ﬁnite number of applications of the above rule is a member of Σ∗.

∗ I A language L over an alphabet Σ is any subset of Σ .

I (In other words, a language is a set of strings).

CSCI 2670 Mathematical Preliminaries Σ∗ and Languages

I Provide a recursive deﬁnition for the language consisting of all strings representing even binary numbers.

I Provide a recursive deﬁnition for the language consisting of all strings representing odd binary numbers.

I Provide a recursive deﬁnition for the language of all bit strings that read the same forwards and backwards.

CSCI 2670 Mathematical Preliminaries Boolean (Propositional) Logic

The language of Boolean logic uses the following alphabet.

I Propositional Variables: p, q, r, etc., with/without subscripts.

I The Logical Connectives (operators): Negation ¬ NOT Conjunction ∧ AND Disjunction ∨ OR Exclusive Disjunction ⊕ XOR Implication → IF...THEN Biimplication ↔ IF AND ONLY IF

I Grouping Symbols ‘(’ and ‘)’.

CSCI 2670 Mathematical Preliminaries Boolean (Propositional) Logic

I Well-formed formulas (wﬀs) of the logic are deﬁned as follows:

I If p is a propositional variable, then p is a wﬀ.

I If p is a wﬀ, then ¬p is a wﬀ.

I If p and q are wﬀs, then the following are all wﬀs:

I (p ∧ q), (p ∨ q), (p ⊕ q), (p → q), (p ↔ q)

I No other strings are wﬀs.

I Note that this is language in the sense described earlier.

I Note also that the wﬀs are deﬁned recursively.

CSCI 2670 Mathematical Preliminaries Boolean (Propositional) Logic

I The propositional variables range over {0, 1}.

I The below tables allow values to be assigned to every wﬀ.

p q (p ∧ q) (p ∨ q) (p ⊕ q) (p → q) (p ↔ q) 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 1 1

p ¬p 1 0 0 1

A formula p is satisﬁable if it is possible to assign 0 and 1 to its propositional variables such that p evaluates to 1.

CSCI 2670 Mathematical Preliminaries Proofs: Terminology

I An axiom is a proposition that is taken as true without question.

I A theorem is a proposition that has been proven to be true (it has been shown to deductively follow from deﬁnitions and axioms).

I A corollary is a proposition that trivially follows from a theorem.

I A lemma is a minor theorem, usually needed in the proof of more important theorems.

I A conjecture is a proposition that has been claimed to be true but not proven.

CSCI 2670 Mathematical Preliminaries Conditional Proofs

I Many mathematical claims are of the form “If P, then Q”.

I One way of proving such a claim is to ﬁrst assume that P is true, and then show that Q must also be true.

I In the proof, you may only use deﬁnitions, axioms, the assumption of p, as well as any other statements that have already been proven.

I This is called a conditional proof or direct proof.

CSCI 2670 Mathematical Preliminaries Proof Strategies: Conditional Proof

Claim: Let A and B be sets. If x ∈ A ∪ B, then x ∈ A ∩ B.

CSCI 2670 Mathematical Preliminaries Proof Strategies: Conditional Proof

Let A and B be sets. If x ∈ A ∪ B, then x ∈ A ∩ B. Proof. Suppose x ∈ A ∪ B. By definition of set complements, x ∈/ A ∪ B. By definition of A ∪ B, x ∈/ A and x ∈/ B. Because of this (again using the definition of set complements), x ∈ A and x ∈ B. According to the definition of set intersection, x ∈ A ∩ B

CSCI 2670 Mathematical Preliminaries Proofs of Biconditionals/Equality

I To prove “P if and only if Q”, perform two proofs:

I the forward direction (Left-to-Right): “If P, then Q”.

I the reverse direction (Right-to-Left): “If Q, then P”.

I E.g. “x ∈ A ∪ B if and only if x ∈ A ∩ B.”

I Show x ∈ A ∪ B implies x ∈ A ∩ B, and

I Show x ∈ A ∩ B implies x ∈ A ∪ B.

I Many proofs for equality are implicitly biconditional proofs.

I E.g., to prove A ∪ B = A ∩ B, you prove the bi-conditional above.

CSCI 2670 Mathematical Preliminaries Existence proofs: Constructive

I Some mathematical propositions claim the existence of something.

I In a constructive proof, the claim is proven by actually exhibiting or constructing the object.

I Claim: There exists an even prime number.

I Proof: It’s 2.

CSCI 2670 Mathematical Preliminaries Existence proofs: Nonconstructive

In a nonconstructive proof, the claim is proven, but no object is constructed or presented.

y Claim: There exist√ irrational numbers x and y such that x is rational. (Consider a = b = 2).

CSCI 2670 Mathematical Preliminaries Existence proofs: Nonconstructive

In a nonconstructive proof, the claim is proven, but no object is constructed or presented.

y Claim: There exist√ irrational numbers x and y such that x is rational. (Consider a = b = 2).

Proof. √ √ Let a = b = 2 (note that 2 is irrational). ab is either rational or it’s not. We consider each case. 1. If ab is rational, then the claim is proven. √ √ √ 2 √ 2 √ 2. If ab is irrational, then let c = ab = 2 . ca = ( 2 ) 2 = √ √ √ 2∗ 2 √ 2 2 = 2 = 2. And so ca is rational. Either way, there exist irrational x and y such that x y is rational. This is a proof by cases. We haven’t ascertained which case obtains, and so we haven’t actually provided a speciﬁc x, y proving the claim.

CSCI 2670 Mathematical Preliminaries Proof by Contradiction

In a proof by contradiction, to prove P, one shows that ¬P implies a contradiction. √ Claim: 2 is irrational.

CSCI 2670 Mathematical Preliminaries Proof by Contradiction

In a proof by contradiction, to prove P, one shows that ¬P implies a contradiction. √ Claim: 2 is irrational.

Proof. √ Suppose for a proof by contradiction that √2 is rational. Then there a exist integers a and b with b 6= 0 such that 2 = b . We can assume without loss of generality that either a or b is odd, since if they are both even ( a = 2c ), we can reduce the fraction further. b 2d √ a a2 2 2 2 Squaring both sides of 2 = b yields 2 = b2 . And so a = 2b . So a is even, which implies that a is even (a = 2k for some integer k). As 2 2 2 2 2 2 2 such, a = (2k) = 4k = 2b , and so 2k = b , which means√ that b and hence b is even. A contradiction! We conclude that 2 is irrational.