Foundations of Computer Science: Logic, Sets, Functions, Induction, and Recursion - Prof. , Exams of Computer Science

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CMPSC 40: Foundations of Computer Science

Key Terms & Results

Peter Cappello

Department of Computer Science

University of California, Santa Barbara

THE FOUNDATIONS: LOGIC & PROOFS

TERMS

proposition: a declarative statement that is true or false, but not both

propositional variable: a variable that represents a proposition

¬p (negation of p): the proposition with truth value opposite to the truth value of p

logical operators: operators used to combine propositions

compound proposition: a proposition constructed by combining propositions using logical operators

p ∨ q (disjunction of p and q): the proposition “p or q,” which is true if and only if at least 1 of p and q is true

p ∧ q (conjunction of p and q): the proposition “p and q,” which is true if and only if both p and q are true

p → q (p implies q): the proposition “if p then q,” which is false if and only if p is true and q is false

p ↔ q (biconditional): the proposition “p if and only if q,” which is true if and only if p and q have the same truth value

p ⊕ q (exclusive or of p and q): the proposition “p XOR q,” which is true when exactly 1 of p and q are true

converse of p → q: q → p

inverse of p → q: ¬p → ¬q

contrapositive of p → q: ¬q → ¬p

tautology: a compound proposition that always is true

contradiction: a compound proposition that always is false

predicate: the part of a sentence that attributes a property to the subject

propositional function: a statement containing 1 or more variables that becomes a proposition when each of its variables is assigned a value or is bound by a quantifier

domain (or universe) of discourse: the set of values a variable in a propositional function may take

∃xP (x) (existential quantification of P (x)): the proposition that is true if and only if there exists an x in the domain such that P (x) is true

∀xP (x) (universal quantification of P (x)): the proposition that is true if and only P (x) is true for every x in the domain

free variable: a variable not bound in a proposition function

bound variable: a variable that is quantified

RESULTS

• The following logical equivalences from Table 6: Double negation: ¬(¬p) ≡ p Commutative: p ∨ q ≡ q ∨ p p ∧ q ≡ q ∧ p Associative: (p ∨ q) ∨ r ≡ p ∨ (q ∨ r) (p ∧ q) ∧ r ≡ p ∧ (q ∧ r) Distributive: p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) DeMorgan’s: ¬(p ∨ q) ≡ ¬p ∧ ¬q ¬(p ∧ q) ≡ ¬p ∨ ¬q
• The following equivalences of implication: p → q ≡ ¬p ∨ q p → q ≡ ¬q → ¬p
• The following equivalences of biconditional: p ↔ q ≡ (p → q) ∧ (q → p) p ↔ q ≡ (p → q) ∧ (¬p → ¬q) p ↔ q ≡ ¬(p ⊕ q)
• DeMorgan’s laws for quantifiers ¬∃P (x) ≡ ∀x¬P (x) ¬∀P (x) ≡ ∃x¬P (x)
• The following rules of inference for propositional logic:

Modus ponens: [p ∧ (p → q)] → q Hypothetical syllogism: [(p → q) ∧ (q → r)] → (p → r) Resolution: [(p ∨ q) ∧ (¬p ∨ r)] → (q ∨ r)

• The rules of inference for quantified statements: Universal instantiation: If ∀xP (x), then P (c) for any c in the domain. Universal generalization: If P (c) for arbitrary c, then ∀xP (x) Existential instantiation: If ∃xP (x), then P (c) for some c in the domain. We however do not know which element in the domain c is. Existential generalization: If P (c) for some c, then ∃xP (x)

SET & FUNCTIONS

TERMS

set: a collection of distinct objects

paradox: a logical inconsistency

element, member of a set: an object in a set

∅ (empty set, null set): the set with no members

universal set: the set containing all objects under consideration

Venn diagram: a graphical representation of a set or sets

S = T (set equality): ∀x(x ∈ S ↔ x ∈ T )

S ⊆ T (S is a subset of T ): ∀x(x ∈ S → x ∈ T )

S ⊂ T (S is a proper subset of T ): S ⊆ T ∧ S 6 = T

finite set: a set with n elements, where n is a natural number

infinite set: a set that is not finite

|S| (the cardinality of S): the number of elements in S

P (S) (the power set of S): {s | s ⊆ S}

A ∪ B (A union B): x ∈ A ∪ B ↔ (x ∈ A ∨ x ∈ B)

A ∩ B (A intersection B): x ∈ A ∩ B ↔ (x ∈ A ∧ x ∈ B)

A − B (A minus B): x ∈ A − B ↔ (x ∈ A ∧ x /∈ B)

A (the complement of A): U − A, where U is the universal set.

A ⊕ B (symmetric difference of A and B): x ∈ A ⊕ B ↔ (x ∈ A ⊕ x ∈ B)

membership table: a table displaying the membership of elements in sets

function from A to B: an assignment such that, ∀a ∈ A, a is assigned to exactly 1 element b ∈ B.

domain of f : the set A, where f is a function from A to B

codomain of f : the set B, where f is a function from A to B

b is the image of a under f: b = f (a)

a is the pre-image of b under f: f (a) = b

THE FUNDAMENTALS: ALGORITHMS & GROWTH OF FUNCTIONS

TERMS

algorithm: a finite sequence of precise instructions for performing a computation or solving a problem.

f (x) is O(g(x)): the fact that |f (x)| ≤ C|g(x)|, for all x > k, for some positive constants C and k.

f (x) is Ω(g(x)): the fact that |f (x)| ≥ C|g(x)|, for all x > k, for some positive constants C and k.

f (x) is Θ(g(x)): the fact that f (x) is O(g(x)) and f (x) is Ω(g(x)).

a | b (a divides b): there is an integer c such that b = ac.

a mod b: the remainder when the integer a is divided by integer b.

a ≡ b (mod m) (a is congruent to b modulo m): m | (a − b).

RESULTS

division algorithm: Let a ∈ Z and d ∈ Z+. Then there are unique q, r ∈ Z with 0 ≤ r < d such that a = dq + r.

INDUCTION & RECURSION

TERMS

the principle of mathematical induction: That the following statement is true:

(P (1) ∧ ∀k[P (k) → P (k + 1)]) → ∀nP (n).

basis step: The proof of P (1) in a proof by mathematical induction of ∀nP (n).

inductive step: The proof of ∀k[P (k) → P (k + 1)] in a proof by mathematical induction of ∀nP (n).

strong induction: That the following statement is true:

(P (1) ∧ ∀k[(P (1) ∧ · · · ∧ P (k)) → P (k + 1)]) → ∀nP (n).

well-ordering property: Every nonempty set of nonnegative integers has a least element.

recursive definition of a function: a definition of a function that specifies an initial set of values and a rule for obtaining values of this function at integers from its values at smaller integers.

recursive definition of a set: a definition of a set that specifies an initial set of elements in the set and a rule for obtaining other elements from those in the set.

structural induction: a technique for proving results about recursively defined sets.

recursive algorithm: an algorithm that proceeds by reducing a problem to the same problem with smaller input.

There are nr^ r-permutations of a set with n elements when repetition is allowed.

There are C(n + r − 1 , r) r-combinations of a set with n elements when repetition is allowed.

There are (^) n 1 !nn 2 !!···nk! permutations of n objects where there are ni indistinguishable objects of type i, for i = 1 , 2 ,... , k.

RECURRENCE RELATIONS

TERMS

recurrence relation: a formula expressing terms of a sequence, except for some initial terms, as a function of 1 or more previous terms of the sequence

initial conditions of a recurrence relation: the values of the terms of a sequence satisfying the recurrence relation before this relation takes effect

divide-and-conquer algorithm: an algorithm that solves a problem recursively by splitting it into a fixed number of smaller problems of the same type

RESULTS

|A ∪ B| = |A| + |B| − |A ∩ B|

RESULTS

1. Let R be an equivalence relation. Then, the following 3 statements are equivalent:
   • aRb.
   • [a]R ∩ [b]R 6 = ∅.
   • [a]R = [b]R.
2. The equivalence classes of an equivalence relation on a set A form a partition of A. Conversely, an equivalence relation can be constructed from any partition so that the equivalence classes are the subsets in the partition.

GRAPHS

TERMS

undirected edge: An edge associated with a set {u, v}, where u and v are vertices.

directed edge: An edge associated with an ordered pair (u, v), where u and v are vertices.

loop: An edge connecting a vertex with itself.

undirected graph: A set of vertices and a set of undirected edges each of which is associated with a set of 1 or 2 of these vertices.

simple graph: An undirected graph with no multiple edges and no loops.

multigraph: An undirected graph that may contain multiple edges but no loops.

directed graph: A set of vertices and a set of directed edges each of which is associated with an ordered pair of vertices.

adjacent: Two vertices are adjacent if there is an edge between them.

incident: An edge is incident to a vertex if the vertex is an endpoint of that edge.

deg(v) (the degree of the vertex v in an undirected graph): The number of edges incident to v with loops counted twice.

deg−(v) (the in-degree of the vertex v in a graph with directed edges): The number of edges with v as their ter- minal vertex.

deg+(v) (the out-degree of the vertex v in a graph with directed edges): The number of edges with v as their initial vertex.

Kn (Complete graph on n vertices): The undirected graph with n vertices where each pair of vertices is connected by an edge.

bipartite graph: A graph with a vertex set that can be partitioned into subsets V 1 and V 2 such that each edge connects a vertex in V 1 and a vertex in V 2.

Km,n (Complete bipartite graph: The graph with a vertex set partitioned into a subset of m vertices and a subset of n vertices such that 2 vertices are connected by an edge if and only if one vertex is in the first subset and the other is in the second subset.

Cn (cycle of size n), n ≥ 3 : The graph with n vertices v 1 , v 2 ,... , vn and edges {v 1 , v 2 }, {v 2 , v 3 },... , {vn− 1 , vn}.

adjacency matrix: A matrix representing a graph using the adjacency of vertices.

incidence matrix: A matrix representing a graph using the incidence of edges of vertices.

circuit: A path of length n ≥ 1 that begins and ends at the same vertex.