What is more surprising is that N (and hence Z) has the same cardinality as the set … /Filter /FlateDecode but "bigger" sets such as $\mathbb{R}$ are called uncountable. Generally, for $n$ finite sets $A_1, A_2, A_3,\cdots, A_n$, we can write, Let $W$, $R$, and $B$, be the number of people with white shirts, red shirts, and black shoes +_R��K*(�qo+r;-���9_\��Q�K�Q�t�t=ZI�)Ƃk����� �v�t{��J{���։���ZCm��)'[�H��=����J�c_�ᣇ�8h�� Then, here is the summary of the available information: >> It turns out we need to distinguish between two types of infinite sets, -�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%�>'���. a proof, we can argue in the following way. Theorem13.1 Thereexistsabijection f :N!Z.Therefore jNj˘jZ. endstream correspondence with natural numbers $\mathbb{N}$. When it comes to infinite sets, we no longer can speak of the number of elements in such a set. According to the de nition, set has cardinality n when there is a sequence but you cannot list the elements in an uncountable set. useful rule: the inclusion-exclusion principle. In mathematics, the cardinality of a set is a measure of the "number of elements" of the set.For example, the set = {,,} contains 3 elements, and therefore has a cardinality of 3. $$B = \{b_1, b_2, b_3, \cdots \}.$$ All of the sets have the same cardinality as the natural numbers ℕ. %���� The second part of the theorem can be proved using the first part. set is countable. where indices $i$ and $j$ belong to some countable sets. 6:38. To be precise, here is the definition. For example, if $A=\{2,4,6,8,10\}$, then $|A|=5$. The Math Sorcerer 19,653 views. endobj S*~����7ׇ�E��bba&�Eo�[email protected]͜9dQ�)ݶ�PSa�a�u��,�nP{|���Jq(jS�z1?m��h�^�aG?c��3>������1p+!��$�R��`V�:�$��� �x�����2���/�d Without loss of generality, we may take $${\displaystyle A}$$ = ℕ = {1, 2, 3,...}, the set of natural numbers. Remember that a function f is a bijection if the following condition are met: 1. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Now, we create a list containing all elements in $A \times B = \{(a_i,b_j) | i,j=1,2,3,\cdots \}$. 18 0 obj xڽXK��D�����M�ꧽ{�� The second part of the theorem can be proved using the first part. Let $A$ be a countable set and $B \subset A$. The cardinality of the denumerable sets is denoted ℵ 0 which is read as "aleph naught" or "aleph null". In elementary set theory, Cantor's theorem is a fundamental result which states that, for any set, the set of all subsets of (the power set of , denoted by ()) has a strictly greater cardinality than itself. Note that another way to solve this problem is using a Venn diagram as shown in Figure 1.11. more concrete, here we provide some useful results that help us prove if a set is countable or not. Becausethebijection f :N!Z matches up Nwith Z,itfollowsthat jj˘j.Wesummarizethiswithatheorem. where $a < b$ is uncountable. so it is an uncountable set. $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$, and any of their subsets are countable. /Filter /FlateDecode respectively. There are many sets that are countably infinite, ℕ, ℤ, 2ℤ, 3ℤ, nℤ, and ℚ. The above arguments can be repeated for any set $C$ in the form of $$|W|=10$$ /Length 1868 refer to Figure 1.16 in Problem 2 to see this pictorially). thus by subtracting it from $|A|+|B|$, we obtain the number of elements in $|A \cup B |$, (you can the inclusion-exclusion principle we obtain. ����{i�V�_�����A|%�v��{&F �B��oA�)QC|*i�[email protected]���$[B��X>�ʏ)+aK6���� -o��� �6� ;�I-#�a�F�*<9���*]����»n�s鿻摞���H���q��ѽ��n�WB_�S����c�ju�A:#�N���/u�,�0ki��2��:����!W�K/��H��'��Ym�R2n�)���2��;Û��&����:��'(��yt�Jzu�*Ĵ�1�&1}�yW7Q���m�M(���Q Ed ���ˀ������C�s� Ӌ��&�Qh��Ou���cJ����>���I6�'�/m��o��m�?R�"o�ͽP�����=�N��֩���&�5��y&���0 �$�YWs��M�ɵ{�ܘ.5Lθ�-� GL��sU 7����>��m�z������lW���)и�$0/�Z�P!�,r��VL�F��C�)�r�j�.F��|���›Y_�p���P׍,�P��d�Oi��5'e��H���-cW_1TRg��LJ��q�(�GC�����7��`Ps�b�\���U7��zM�d*1ɑ�]qV(�&3�&ޛtǸ"�^��6��Q|��|��_#�T� Before discussing Consider a set $A$. If A is a finite set, then | B | ≤ | A | < ∞, thus B is countable. where one type is significantly "larger" than the other. The above theorems confirm that sets such as $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$ and their If $A_1, A_2,\cdots$ is a list of countable sets, then the set $\bigcup_{i} A_i=A_1 \cup A_2 \cup A_3\cdots$