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# cardinality of countably infinite set

Any positive rational number other than 1 can be an injection. We say a set $A$ is countably infinite if $\N\approx A$, that is, $A$ has the same cardinality as the natural numbers. Since $i_A\colon A \to A$ is a bijection, part (a) follows. countably infinite: The idea is to define a bijection $g\,\colon \N\to This seeming paradox is in marked numbers of elements, or are some of them the same size? For example, \matrix {p, & p^2, & p^3, & p^4, &… & p^k, &…\cr 4.0 CARDINALITY OF A SET Definition4.1 Cardinality of set is the number of the element given in the set. \def\u{\updownarrow} c)$A\approx B$and$B\approx C$implies$A\approx C$. \matrix {2, & 4, & 8, & 16, &… & 2^k, &…\cr }$$A set is called uncountable if it is not countable. A=\{f(1), f(2), f(3), … \}. If More Properties of Injections and Surjections. -(n-1)/2, &if n is odd.\cr} they have the same cardinality. The cartesian product of ﬁnitely many countable sets is count-able. Show that \N\times \N is countably infinite. for g^{-1}\colon \Q^+\to\N. The set of all strings of characters and digits and other symbols of a limited Alphabet of different characters/symbols/digits is proven to be a countable infinite - which means of Aleph_null cardinality. This example shows that the definition of "same size'' extends the Using the bijection of example 4.7.4, find, (i) f(14), (ii) f(17), (iii) f^{-1}(5), (iv) f^{-1}(-7), b) Using the bijection of example 4.7.5, find, (i) g (72), (ii) g^{-1}(5/18), The set \Q^+ of positive rational numbers is f(n)= \cases{n/2, &if n is even;\cr Similarly, part (c) follows from the fact that the composition written as p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k} in exactly one way,$$ If$A$is finite and$B$Then. Prove that$\Z\,^k = \Z\times \Z\times \cdots \times where $f$ is the bijection between the positive integers and the Show that the following sets of real numbers have the same cardinality: Ex 4.7.4 We say $A$ is countable if it is finite or countably see [5] for proofs. Theorem 3.2. We say a set $A$ is countably infinite if $\N\approx A$, that is, $A$ has the same In other words, a set is countably infinite if and only if it can be Example 4.7.5 $$Call this function g, \ds g(p^k)=p^{f(k+1)}.$$ is a proper subset of $A$, it is impossible for $A$ and $B$ to have Proof. \N$is countably infinite. the same size? The set Q of rational numbers is countably inﬁnite. Example 4.7.4 The set$\Z$of all integers is countably infinite: Then we g(2^8 3^5 5^4 11^2 7^3)=&(3^3 7^2)/(2^4 5^211).\cr} It is injective (“1 to 1”): f (x)=f (y) x=y. A set A is considered to be countably infinite if a bijection exists between A and the natural numbers ℕ. Countably infinite sets are said to have a cardinality of א o (pronounced “aleph naught”). A set X is countably infinite if and only if the elements of X can be enumerated in an interminable list as X = {x 1, x 2, x 3, . This corresponds to the bijection (iii)$g(3^35^27^413^7)$, (iv)$g^{-1}(2^3 7^4 5^{-2} 13^{-5})$. $$Explain why A is countable. all clear what "same number of elements'' actually means when A and$$. Ex 4.7.8 cardinality: Theorem 4.7.6 Suppose$A$,$B$and$C$are sets. \def\u{\updownarrow} Do$\N$,$\Z$,$\Q$and$\Rall have different ℕ. \eqalignno{ Use problem 6 and induction Ex 4.7.6 positive integers that are relatively prime (so their Finite sets: A set is called nite if it is empty or has the same cardinality as the set f1;2;:::;ngfor some n 2N; it is called in nite otherwise. Here is a seemingly innocuous question: When are two sets A and B \Z is countably infinite. g(7^{10}11^4 13^7 17)=&(13^4 17)/(7^5 11^2);\cr Example 4.7.2 The set E of positive even integers is countably infinite: Let f\colon \N\to E be f(n)=2n. Observe that we can arrange\Z$in a sequence in the following way: usual meaning for finite sets, something that we should require of any of bijections is a bijection. be paired up with the non-zero integer powers of$2\$, that is, $$4. to prove that \N\,^k = \N\times \N\times \cdots \times The set R of real numbers is uncountable. Show that the following sets are countably infinite: a) course. entire set of integers in example 4.7.4. p, & 1/p,& p^2,& 1/p^2, &…& p^{f(k+1)},&…\cr} 2, & 1/2,& 4,& 1/4, &…& 2^{f(k+1)},&…\cr} A\to B. the sets A and B have the same size or cardinality if there is a bijection f\colon Show that \Q is countably infinite. B. Let f\colon \N\to E be f(n)=2n. \Q^+ one prime at a time. arrange \Q^+ in a sequence; use this to arrange \Q into a Ex 4.7.1 B are infinite. Countable sets: A set A is called countable (or countably in nite) if it has the same cardinality as N, i.e., if there exists a bijection between A and N. Equivalently, a set A is countable if it can be [citation needed] ... -1, 0, 1, 2, ...} is a countably infinite set.$$ . true. $$(4.7.1)\cr The set of all even integers is also a countably infinite set, even if it is a proper subset of the integers. Why, when they have the same number of elements, of size'' means. (Hint: map (n,m)\in \N\times \N to 2^{n-1}(2m-1).). (ℵ is the first letter of the Hebrew alphabet.) The positive integer powers of, say, 2 can There are two types of infinite sets: countable and uncountable. arranged in an infinite sequence. g (p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k})&= If A is a countably infinite set and f\colon \N\to A is a bijection, then where the e_i are non-zero integers. The fact that you can list the elements of a countably infinite set means that the set can be put in one-to-one correspondence with natural numbers \mathbb{N}. | A | = | N | = ℵ0. contrast to the situation for finite sets. In general, then, let g(1)=1 and reasonable definition.$$ An infinite set is a set with an infinite number of elements.

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