# proving cardinality of infinite sets

There are many sets that are countably infinite, ℕ, ℤ, 2ℤ, 3ℤ, nℤ, and ℚ. then talk about infinite sets. stream If set A is countably infinite, then | A | = | N |. list its elements: $A_i=\{a_{i1},a_{i2},\cdots\}$. thus by subtracting it from $|A|+|B|$, we obtain the number of elements in $|A \cup B |$, (you can Figure 1.13 shows one possible ordering. The cardinality of a set is denoted by $|A|$. where $a < b$ is uncountable. %���� subsets are countable. Hence these sets have the same cardinality. | A | = | N | = ℵ0. To provide endstream Without loss of generality, we may take $$A$$ = ℕ = {1, 2, 3,...}, the set of natural numbers. where indices $i$ and $j$ belong to some countable sets. If $A$ and $B$ are countable, then $A \times B$ is also countable. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between the different types of infinity, and to perform arithmetic on them. $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$, and any of their subsets are countable. you can never provide a list in the form of $\{a_1, a_2, a_3,\cdots\}$ that contains all the What is more surprising is that N (and hence Z) has the same cardinality as the set … $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$. Now, we create a list containing all elements in $A \times B = \{(a_i,b_j) | i,j=1,2,3,\cdots \}$. Also, it is reasonable to assume that $W$ and $R$ are disjoint, $|W \cap R|=0$. should also be countable, so a subset of a countable set should be countable as well. If A is a finite set, then | B | ≤ | A | < ∞, thus B is countable. useful rule: the inclusion-exclusion principle. 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”). Let $A$ be a countable set and $B \subset A$. Before discussing However, to make the argument If A is countably infinite, then we can list the elements in A, then by removing the elements in the list that are not in B, we can obtain a list for B, thus B is countable. If $B \subset A$ and $A$ is countable, by the first part of the theorem $B$ is also a countable For example, you can write. Cardinality of inﬁnite sets The cardinality |A| of a ﬁnite set A is simply the number of elements in it. countable, we can write The proof of this theorem is very similar to the previous theorem. And n (A) = 7. The difference between the two types is For example, the absolute value of a real number measures its size in terms of how far it is from zero on the number line. For example, let A = { -2, 0, 3, 7, 9, 11, 13 } Here, n(A) stands for cardinality of the set A. refer to Figure 1.16 in Problem 2 to see this pictorially). /Filter /FlateDecode then by removing the elements in the list that are not in $B$, we can obtain a list for $B$, That is, there are 7 elements in the given set … >> Thus, we have. A set A is countably infinite if and only if set A has the same cardinality as N (the natural numbers). If $A_1, A_2,\cdots$ is a list of countable sets, then the set $\bigcup_{i} A_i=A_1 \cup A_2 \cup A_3\cdots$ The second part of the theorem can be proved using the first part. endobj << The above arguments can be repeated for any set $C$ in the form of .�i���丳a� r��s|��n���ߌx5̹�d�:��e��@�;�_�4-���@��xi�Z�۫'���M���u�pF��>\��Z���Y0��W��mG�� Cardinality of a set is a measure of the number of elements in the set. (Also known as countably infinite.) For example, a consequence of this is that the set of rational numbers $\mathbb{Q}$ is countable. set is countable. /Length 1933 stream The Infinite Looper 48,403 views. $$\>\>\>\>\>\>\>+\sum_{i < j < k}\left|A_i\cap A_j\cap A_k\right|-\ \cdots\ + \left(-1\right)^{n+1} \left|A_1\cap\cdots\cap A_n\right|.$$, $= |W| + |R| + |B|- |W \cap R| - |W \cap B| - |R \cap B| + |W \cap R \cap B|$. number of elements in $A$. -�ޗ�8Y��He�����`��S���}$�a��SdV���$6��M� i��sЇ�K�mI 8���cS�}����h����DTq�#��w�yD>�ۨQ��e��,f�͋ խ�c[[����0����4bT�EAF�Eo�0kW�m�u� i�S{���I%GbP����I%�>'���. Let us examine the proof for the specific case when $$A$$ is countably infinite. For two finite sets $A$ and $B$, we have One may be tempted to say, in analogy with finite sets, that all thus $B$ is countable. set whose elements are obtained by multiplying each element of Z by k.) The function f : N !Z de ned by f(n) = ( 1)nbn=2cis a 1-1 corre-spondence between the set of natural numbers and the set of integers (prove it!).

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