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.