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Topic: Equinumerous

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	Bloomsbury.com - Research centre (Site not responding. Last check: 2007-10-09)
		Two sets are 'equinumerous' (equal in number) if there is a mapping between them which is a bijection, that is, a function where the two non-equal elements in the first set have non-equal images, and where every element in the second set has an element which the function maps to it.
		For example, the sets 1,2 and 5,7 are equinumerous, because the function mapping 1 to 5 and 2 to 7 is a bijection.
		Many kinds of numbers are countable, such as the integers and the rational numbers, while others are not (the way that the real numbers are shown to be uncountable is in algebraic numbers).
	www.bloomsburymagazine.com /ARC/detail.asp?EntryID=102274&bid=2 (423 words)

	cardinality
		(def-theorem equinumerous-is-symmetric "forall(a:sets[ind_1],b:sets[ind_2], (a equinumerous b) implies (b equinumerous a))" (theory pure-generic-theory-2) (usages transportable-macete) (proof (direct-and-antecedent-inference-strategy (instantiate-existential ("inverse{f}")) insistent-direct-inference-strategy (apply-macete-with-minor-premises dom-of-inverse) (apply-macete-with-minor-premises injective-iff-injective-on-domain) simplify (apply-macete-with-minor-premises ran-of-inverse) (instantiate-theorem inverse-is-injective ("f")) (backchain "with(f:[ind_1,ind_2],injective_q{inverse{f}})"))))
		(def-theorem equinumerous-is-transitive "forall(a:sets[ind_1],b:sets[ind_2],c:sets[ind_3], (a equinumerous b) and (b equinumerous c) implies (a equinumerous c))" (theory pure-generic-theory-3) (usages transportable-macete) (proof (direct-and-antecedent-inference-strategy (instantiate-existential ("f_$0 oo f")) insistent-direct-inference-strategy (apply-macete-with-minor-premises domain-composition) (apply-macete-with-minor-premises range-composition) (cut-with-single-formula "injective_q{f}") (cut-with-single-formula "injective_q{f_$0 oo f}") (backchain "with(f:[ind_1,ind_2],f_$0:[ind_2,ind_3], injective_q{f_$0 oo f})") (apply-macete-with-minor-premises injective-composition) simplify (apply-macete-with-minor-premises injective-iff-injective-on-domain) simplify)))
		(def-theorem equinumerous-to-empty-indic "forall(a:sets[ind_1], (a equinumerous empty_indic{ind_2}) iff a=empty_indic{ind_1})" (theory pure-generic-theory-2) (usages transportable-macete) (proof (direct-inference-strategy (apply-macete-with-minor-premises rev%embeds-in-empty-indic) (apply-macete-with-minor-premises equinumerous-implies-embeds) (force-substitution "a" "empty_indic{ind_1}" (0)) (instantiate-existential ("lambda(x:ind_1,?ind_2)")) insistent-direct-inference-strategy extensionality direct-inference simplify extensionality direct-inference simplify simplify (beta-reduce-antecedent "with(p,q,r:prop, p and q and r)"))))
	imps.mcmaster.ca /theories/cardinality/cardinality.html (327 words)

	[No title]
		The first is show that the whole numbers are equinumerous to the irrational numbers represented by the positive square roots.
		From here it can be shown that the natural numbers are equinumerous to all rational numbers by mapping the even natural numbers to the positive rationals, the odd natural numbers to the negative rationals, and 0 to 0.
		Therefore, the natural numbers are equinumerous to the irrational numbers represented by positive whole number roots of rational numbers.
	www.public.iastate.edu /~joefish/proof.doc (382 words)

	Cardinal assignment . Axiom of choice . Gottlob Frege (Site not responding. Last check: 2007-10-09)
		The concepts are developed by defining equinumerous equinumerosity in terms of functions and the concepts of one-to-one and onto injectivity and surjectivity ; this gives us an pseudo-ordering relation :A \leq_c B\quad \iff\quad \exists f f : A \to B\ \mathrm on the whole universe by size.
		It is not a true ordering because the trichotomy trichotomy law need not hold: if both A \leq_c B and B \leq_c A, it is true by the Cantor-Bernstein-Schroeder theorem that A =_c B i.e.
		A and B are equinumerous, but they do not have to be literally equal; that at least one case holds turns out to be equivalent to the Axiom of choice.
	www.uk.kunsimuna.net /Cardinal_assignment_UK_317133_tb (472 words)

	Logic Discussion Board :: View topic - Are the amounts of prime numbers and numbers equal?
		A is equinumerous to B if and only if there exists a 1-1 function from A onto B. By the way, the relation of equinumerosity is symmetric.
		That is, if A is equinumerous to B, then B is equinumerous to A, since if there is a 1-1 function from A onto B, then there is a 1-1 function from B onto A (in particular, the inverse of the function from A onto B will do).
		In this sense of equinumerosity, the set of primes is equinumerous to the set of natural numbers.
	www.christianlogic.com /forums/topic657.html&start=6&sid=74af216b6f0b1160f0d993a6edb62ad5 (564 words)

	Frege's Derivation of Hume's Principle in the Grundlagen: A Supplement to Frege's Logic, Theorem, and Foundations for ...
		Then, by the definitions of ‘the number of Ps’ and ‘the number of Qs’, we know that the extension of the concept equinumerous to P is identical with the extension of the concept equinumerous to Q. But it is a fact about equinumerosity that P is equinumerous to P.
		So, by definition, we have to show that the extension of the concept equinumerous to P is identical to the extension of the concept equinumerous to Q.
		So we pick an arbitrary concept S and show that S is a member of the extension of the concept equinumerous to P iff S is a member of the extension of the concept equinumerous to Q.
	www.seop.leeds.ac.uk /entries/frege-logic/Gl-Hume.html (521 words)

	natural religion > glossary > Cantor's diagonal proof (Site not responding. Last check: 2007-10-09)
		By countable, we mean equinumerous with the (infinite) set of natural numbers generated by Peano's axioms.
		At first glance, we might feel that the set of fractional numbers is much bigger than the set of natural numbers, since there are many fractions between each pair of natural numbers.
		Closer examination, however, reveals that the set of natural numbers and the set of fractions are equinumerous.
	www.naturaltheology.net /Glossary/cantorDiagonal.html (541 words)

	Did a Number Conquer Gaul
		Two concepts are equinumerous if and only if there is some one-to-one relationship between all the objects falling under one concept and all the objects falling under the other.
		The number (referred to by the symbol) 0 is the extension of the concept “equinumerous to the concept of not being self-identical”.
		Additionally, it is possible to define the successor of 0 as the extension of the concept “equinumerous to the concept of being identical with 0”.
	www.gmalivuk.com /otherstuff/wint03/phil418a.htm (3302 words)

	[No title]
		A set is finite if it is equinumerous with the set {1, 2,..., n}, for a given natural number n.
		If A and {1,..., n} are equinumerous, then the "cardinality" of A is n.
		A set is "countably infinite" if it is equinumerous with N, and "countable" if it is finite or countably infinite.
	ranger.uta.edu /~cook/tcs/l3.html (821 words)

	[No title]
		Any set which is equinumerous with such a set is called finite.
		to be equinumerous) by presenting an arrangement in which their elements are put into 'one-to-one correspondance' [see the section on Functions later].
		we may conclude that the sets are equinumerous and that therefore the set of even numbers is countable.
	www.soi.city.ac.uk /~bernie/dm2000/pg/Week1/c16.doc (578 words)

	Infinite Set Equinumerous (Site not responding. Last check: 2007-10-09)
		...every infinite set is equinumerous with an aleph number.
		By countable, we mean equinumerous with the (infinite) set of...
		...if two infinite sets are the same "size" (equinumerous) by seeking to find a one-to-one match-up between the elements of each set.
	www.wildlife-expedition.com /Le0P0-more/Infinite-Set-Equinumerous.html (234 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		The basic idea is that of equinumerosity: two sets $X$ and $Y$ are said to be {\em equinumerous} (written $X \sim Y$) if $\exists f:X \leftrightarrow Y$ such that $f$ is a bijection.
		Notice that this leaves us in an odd situation: the set $\cal Z$ of all integers and the set $\cal E$ of even integers, for instance, are equinumerous.
		There are many ways to do this, but perhaps the simplest is to simply assert that infinite sets also have sizes or cardinalities, and that if two infinite sets are equinumerous then their cardinalities are the same.
	www.cs.umd.edu /class/spring2000/cmsc251/notes/cards.text (1467 words)

	Metamath Proof Explorer - carden
		Description: Two sets are equinumerous iff their cardinal numbers are equal.
		This important theorem expresses the essential concept behind "cardinality" or "size".
		The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [
	us.metamath.org /mpegif/carden.html (119 words)

	Logical Constructions
		Examples he cited were the Frege/Russell definition of numbers as classes of equinumerous classes, the theory of definite descriptions, the construction of matter from sense data, and several others.
		Russell's logicist program could not rest content with postulates for the fundamental objects of mathematics such as the Peano Axioms for the natural numbers.
		Instead numbers were to be defined as classes of equinumerous classes.
	plato.stanford.edu /entries/logical-construction (2377 words)

	Logicism I: Frege
		Frege argues that we can understand the number of things falling under a concept F by way of the more general concept of being equinumerous.
		Thus, we use the idea that (1) the extension of the concept F is equinumerous with the extension of the concept G, to make sense of the idea that (2) the number of Fs is the same as the number of Gs.
		Frege says, "The Number that belongs to the concept F is the extension of the concept 'equinumerous to the concept F'"
	www.oswego.edu /~delancey/309_DIR/LLT_LECTURES/5_frege_out.html (1653 words)

	Introduction to the
		A. Proving that "Equinumerous" satisfies the third property, Transitivity, is left as the next exercise.
		A set that is Equinumerous with the set of Natural numbers is said to be Denumerable.
		Thus A is less than or Equinumerous with B if and only if A is Equinumerous with a subset of B. Sometimes this statement is used as a definition of Less than or Equinumerous and our definition earlier is derived as a consequence.
	www.math.toronto.edu /jkorman/Math246Y/transcard.htm (9604 words)

	Gottlob Frege
		With this notion of equinumerosity, Frege defined ‘the number of the concept F’ (or, more informally, ‘the number of Fs’) to be the extension or set of all concepts that are equinumerous with F (1884, §68).
		For example, the number of the concept author of Principia Mathematica is the extension of all concepts that are equinumerous to that concept.
		Frege thereby identified the number 0 as the class of all concepts under which nothing falls, since that is the class of concepts equinumerous with the concept not being self-identical.
	plato.stanford.edu /entries/frege (10319 words)

	The permutation classes equinumerous to the Smooth class (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		The permutation classes equinumerous to the Smooth class (ResearchIndex)
		The permutation classes equinumerous to the Smooth class (1998)
		Abstract: We determine all permutation classes defined by pattern avoidance which are equinumerous to the class of permutations whose Schubert variety is smooth.
	citeseer.ist.psu.edu /59919.html (324 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		DEFINITION: Two sets A and B are called *equinumerous* or are said to have the same cardinality if there is a bijection f : A-->B. [bijection or 1-1 correspondence means that f is 1-1 and onto.
		It's a bit counterintuitive to say that these two sets are equinumerous since the second is a proper subset of the first.
		DEFINITION: A set is finite if there is a bijection between that set and {1, 2,..., n} for some number n.
	www.columbia.edu /~mpj9/4236/card.txt (537 words)

	Frege's ‘Derivation’ of Hume's Principle in the Grundgesetze: A Supplement to Frege's Logic, Theorem, and ...
		(What follows is an adaptation and simplification of the strategy Frege outlines in Gg I, §34ff.) Instead of defining the number of Fs as the extension consisting of all those first-order concepts that are equinumerous to F, he defined it as the extension consisting of all the extensions of concepts equinumerous to F.
		This Lemma tells us that an extension such as εG will be a member of #F just in case G is equinumerous to F.
		Clearly, since F is equinumerous to itself, it follows that #F contains εF as a member.
	setis.library.usyd.edu.au /stanford/archives/spr2004/entries/frege-logic/Gg-Hume.html (368 words)

	equinumeroso - equinumerous (Spanish to English translation glossary) Mathematics,Mathematics & Statistics,
		Could somebody please help me with the translation of "equinumeroso".
		Dos conjuntos abstractos pueden estar en correspondencia uno-a-uno entre ellos y sin embargo esto no significa que sean "equinumerosos" porque bien puede ocurrir que uno de ellos esté contenido como subconjunto propio del otro.
		Nikki Graham: yes, equinumerous, but please put your answer in the correct place
	www.proz.com /kudoz/540893 (226 words)

	Equinumerosity (Site not responding. Last check: 2007-10-09)
		Two sets A and B are equinumerous if they have the same cardinality, i.e., if there exists a bijection f : A → B.
		The study of cardinality is often called equinumerosity.
		In Set, the category of all sets with functions as morphisms, an isomorphism between two sets is precisely a bijection, and two sets are equinumerous precisely if they are isomorphic in this category.
	www.worldhistory.com /wiki/E/Equinumerosity.htm (152 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		equinumerous to the set of all natural numbers.
		numbers is not equinumerous to the set of all infinite sequences of
		the power set of \omega is NOT equinumerous to \omega.
	math.uci.edu /~mzeman/M151 (646 words)

	My homepage
		Show that the set of all natural numbers and the set of all odd numbers are equinumerous.
		X of natural numbers there is a prime number
		Show that the set of all prime numbers and the set of all natural numbers are equinumerous.
	www.contrib.andrew.cmu.edu /~yimuy/teaching/801102004s/0618.htm (150 words)

	Countable Ordinals (Site not responding. Last check: 2007-10-09)
		tau equinumerous a === tau in S some bijection p:tau -= a R = { (x,y)
		Otherwise \/S + 1 equinumerous a \/S in \/S + 1 in S; \/S in \/S which cannot be =Let X be a set and R be a well-ordering of X. The (order) =type of the well-ordered set
		In particular would you assist showing the next part: To show tau and X equinumerous** one needs f is injection.
	www.forum-one.org /new-5971498-4346.html (6101 words)

	Frege's `Derivation' of Hume's Principle in Gg (Site not responding. Last check: 2007-10-09)
		Instead of writing out this lengthy expression being an x which is an extension of a concept equinumerous to F, let us abbreviate our
		will be a member of #F just in case G is equinumerous to F.
		Clearly, since F is equinumerous to itself, it follows that #F contains
	www.science.uva.nl /~seop/archives/win1999/entries/frege-logic/Gg-Hume.html (279 words)

	CmSc 365 Theory of Computation
		Finite sets - finite number of elements - can be counted
		Sets A and B are equinumerous if there is a bijection f : A → B
		A set is countably infinite if it is equinumerous with N (the set of natural numbers)
	www.simpson.edu /~sinapova/cmsc365-02/L02-Countability.htm (791 words)

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