Theory of Computation Quiz 1

Biconditional

(P->Q) ^ (Q->P) is often abbreviated as P<->Q and is read, P iff Q.

Principle of Mathematical Induction

Let P(n) be a statement where n is a natural number. Then if {P(0) is true and ((for all k) P(k) is true -> P(k+1) is true)} -> P(n) is true for all n.

Denumerable

An infinite set A is called denumerable iff it is either finite or countable. Otherwise it is uncountable.

Statement (Proposition)

Any sentence for which truth or falsity can be determined.

Universal and Existential Quantifiers

Certain operators indicate to us how we may select items from the set of meanings

Set

Collection of objects

Universally Quanitifed

For all x in the set of meanings P(x) is true. Indicates that the truth set of P(c) consists of all objects in the set of menaings for x.

Function

Form A to B is a relation from A to B that meets additional criteria. The set f subset AxB is a function of Domain(f) = A and whenever (x,y) and (x,z) are in f. Then y=z. That means every element x of A had uniquely determined y, such that (x,y) € f.

Subjective (onto)

Function f is said to be subjective if whenever y€B, there exists some x€A, for which f(x)=y.

Injective (one to one)

Function f:A->B is injective if whenever (x,y) € f and (z,y) € f, then x=z.

Equivalent

If P and Q are statements, we say that P and Q are equivalent if they have the same value.

Negation

If P is a statement, its negation is denoted not P. Therefore if P is true, then not P is false.

Bijection (one to one correspondence)

If f is both one to one and onto then it is a bijection.

Inverse relation

If ro is a subset of AxB then set ro ^ -1 = {(b,a) | (a,b) € ro } is a subset of BxA.

Partition

Let A be a non empty set. A collection pi of nonempty subsets of A is a partition of A if the following holds: 1. If B subset A and C subset A, then either B=C or B intersects C = null. 2. Pi is the union for B.

Indexed Family of Sets

Let I be some set. If for each ro € I, we have that A ro is a set, then {A ro | ro € I}.

Cardinality

Of A, |A| is thr number of elements in A.

Partition pi

Of a nonempty set A defines an equivalence relation on A

Power set

Of set A is denoted as 2^A, collection of all subsets of A.

Cartesian Product

Of sets A and B, AxB.

Truth Set

Of the open sentence is the set of objects in the set of meanings for which the open sentence becomes true.

Set of Meanings

Of the variable in an open sentence is the collectionof all objects which can be substituted for that variable.

Basis

P(0) is true and is the basis of induction.

Implication (Conditional)

Proposition P->Q is the implication and had the truth table: whenever T -> F, then F.

Relation

Relation Ro from Set A and set B is a subset of AxB.

Partial Function

Relation f subsets AxB, Dom(f) subsets A and (x,y) € f and (x,z) € f implies y=z.

Subset

Set A is a subset of set B iff: For all ( X € A -> x € B)

Finite

Set A is finite if there exists a one to one correspondenxe brtwern the eleemnts Of A and the elements of Nk = {1,2,3...k}

Countable

Set B is countable iff there exists a one to one correspondence between the elements of B and elements of N.

Empty (Null) Set

Set with no memebers

Open Sentence (Propositional Function)

Statement involving a variable.

Taurology

Statement is always true.

Contradiction

Statement that is always false.

Relation Ro on Set A

Subset of AxA.

Conjunction

The conjunction of propositions P and Q is deonted P^Q, read as P and Q. True when both are true.

Contrapositive

The contrapositive of P->Q is notQ->notP.

Converse

The converse of P->Q is Q->P.

Disjunction

The disjunction of propositions P and Q is denoted P\/Q, read as P or Q. True whenever at least one is true.

Equivalence class

The set bracket [x]

Existentially Quantified

There exists an x in the set of meanings for which P(x) is true. Indicates that somewhere in the set or meanings is a value that when subsituted for x makes P(x) true.

Equal

Two sets are equal iff they have the same elements.

Equivalence Relation

When relation ro is reflexive, symmetric, and transitive.

Strong Induction

{P(0) is true and ((for all k) ((for all r) P(r) is true -> P(r+1) is true))} -> P(n) is true for all n.