R[ȱ �3&��)���f��!%GV'�%�H���S��o�l��ֹ���w�}xU��c���}�Z{%�`B�� However, to make the argument case the set is said to be countably infinite. Cardinality of a set is a measure of the number of elements in the set. �L�`2��T�bg���H���g�-.n?�����|������xw���.�b6����,��,�fr��X��}ޖ�]uX��ՙ]�q�3����S�Œ��P7���W?s��c[u-���hEK�c��^�e�\�� Consider a set $A$. $$\biggl|\bigcup_{i=1}^n A_i\biggr|=\sum_{i=1}^n\left|A_i\right|-\sum_{i < j}\left|A_i\cap A_j\right|$$ | A | = | N | = ℵ0. Thus U is both uncountable and countably infinite, a contradiction. If $A$ and $B$ are countable, then $A \times B$ is also countable. where one type is significantly "larger" than the other. If $A$ is countably infinite, then we can list the elements in $A$, }أ��-W�������"�C��� �00�!�U嚄l�'}�� -F�NQ gvPC���S�:|����ա���ʛ������v���|�Z���uo�2�a`ynް�K��gS��v�������*��P~Ē�����&63 If $A$ has only a finite number of elements, its cardinality is simply the If $A_1, A_2,\cdots$ is a list of countable sets, then the set $\bigcup_{i} A_i=A_1 \cup A_2 \cup A_3\cdots$ It turns out we need to distinguish between two types of infinite sets, ����RJ�IR�� $$B = \{b_1, b_2, b_3, \cdots \}.$$ is also countable. refer to Figure 1.16 in Problem 2 to see this pictorially). Sets such as $\mathbb{N}$ and $\mathbb{Z}$ are called countable, thus $B$ is countable. Any set containing an interval on the real line such as $[a,b], (a,b], [a,b),$ or $(a,b)$, Definition13.1settlestheissue. Theorem13.1 Thereexistsabijection f :N!Z.Therefore jNj˘jZ. Thus, we have. /Length 1868 Assume $B$ is uncountable. should also be countable, so a subset of a countable set should be countable as well. $$\mathbb{Q}=\bigcup_{i \in \mathbb{Z}} \bigcup_{j \in \mathbb{N}} \{ \frac{i}{j} \}.$$. The above theorems confirm that sets such as $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$ and their If you are less interested in proofs, you may decide to skip them. Becausethebijection f :N!Z matches up Nwith Z,itfollowsthat jj˘j.Wesummarizethiswithatheorem. set which is a contradiction. This is because we can write However, as we mentioned, intervals in $\mathbb{R}$ are uncountable. is concerned, this guideline should be sufficient for most cases. Before discussing We can extend the same idea to three or more sets. endobj And n (A) = 7. Some sets that are not countable include ℝ, the set of real numbers between 0 and 1, and ℂ. If set A is countably infinite, then | A | = | N |. << In particular, one type is called countable, On the other hand, you cannot list the elements in $\mathbb{R}$, Thus by applying $$A = \{a_1, a_2, a_3, \cdots \},$$ 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!). One may be tempted to say, in analogy with finite sets, that all ?�iOQ���i�ʫ�.��_���׉!ә�. For example, if $A=\{2,4,6,8,10\}$, then $|A|=5$. $$|W|=10$$ -�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%�>'���. All of the sets have the same cardinality as the natural numbers ℕ. $$\>\>\>\>\>\>\>+\sum_{i < j < k}\left|A_i\cap A_j\cap A_k\right|-\ \cdots\ + \left(-1\right)^{n+1} \left|A_1\cap\cdots\cap A_n\right|.$$, $= |W| + |R| + |B|- |W \cap R| - |W \cap B| - |R \cap B| + |W \cap R \cap B|$. $$|A \cup B |=|A|+|B|-|A \cap B|.$$ JHY!Ǭ4�5�8s�-�����D��~�)�X�`ŐN/�*��u��yߺ��6�ԏ@�Z����R�9�pf�C�`Te�n�K�T��`X*��� ��T� F,���˙�� �M�zϞ��=Fr������JLL`�W��EO��& $$|R|=8$$ S*~����7ׇ�E��bba&�Eo�oRB@3a͜9dQ�)ݶ�PSa�a�u��,�nP{|���Jq(jS�z1?m��h�^�aG?c��3>������1p+!��$�R��`V�:�$��� �x�����2���/�d 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. ����{i�V�_�����A|%�v��{&F �B��oA�)QC|*i�P@c���$[B��X>�ʏ)+aK6���� -o��� �6� ;�I-#�a�F�*<9���*]����»n�s鿻摞���H���q��ѽ��n�WB_�S����c�ju�A:#�N���/u�,�0ki��2��:����!W�K/��H��'��Ym�R2n�)���2��;Û��&����:��'(��yt�Jzu�*Ĵ�1�&1}�yW7Q���m�M(���Q Ed ���ˀ������C�s� Ӌ��&�Qh��Ou���cJ����>���I6�'�/m��o��m�?R�"o�ͽP�����=�N��֩���&�5��y&���0 �$�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� then talk about infinite sets. >> thus $B$ is countable. The function f : … SetswithEqualCardinalities 219 N because Z has all the negative integers as well as the positive ones. $$|W \cup B \cup R|=21.$$ Since each $A_i$ is countable we can How To Tell If Subwoofer Is Out Of Phase, Ph Of Avocado Oil, Fried Shrimp Roll Recipe, Fruit Grading System, Hairy Bikers Hazelnut Brownies, Best Mattress At Costco, Screw On Mic Cable, " />
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proving cardinality of infinite sets

For example, a consequence of this is that the set of rational numbers $\mathbb{Q}$ is countable. +_R��K*(�qo+r;-���9_\��Q�K�Q�t�t=ZI�)Ƃk����� �v�t{��J{���։���ZCm��)'[�H��=����J�c_�ᣇ�8h�� $$|R \cap B|=3$$ stream 18 0 obj Now, we create a list containing all elements in $A \times B = \{(a_i,b_j) | i,j=1,2,3,\cdots \}$. The intuition behind this theorem is the following: If a set is countable, then any "smaller" set The cardinality of a set is denoted by $|A|$. $\mathbb{Z}=\{0,1,-1,2,-2,3,-3,\cdots\}$. correspondence with natural numbers $\mathbb{N}$. 7.3. p���02[/5�ph)zr+��ŀ�eT%��\ �U��/t��E+>R[ȱ �3&��)���f��!%GV'�%�H���S��o�l��ֹ���w�}xU��c���}�Z{%�`B�� However, to make the argument case the set is said to be countably infinite. Cardinality of a set is a measure of the number of elements in the set. �L�`2��T�bg���H���g�-.n?�����|������xw���.�b6����,��,�fr��X��}ޖ�]uX��ՙ]�q�3����S�Œ��P7���W?s��c[u-���hEK�c��^�e�\�� Consider a set $A$. $$\biggl|\bigcup_{i=1}^n A_i\biggr|=\sum_{i=1}^n\left|A_i\right|-\sum_{i < j}\left|A_i\cap A_j\right|$$ | A | = | N | = ℵ0. Thus U is both uncountable and countably infinite, a contradiction. If $A$ and $B$ are countable, then $A \times B$ is also countable. where one type is significantly "larger" than the other. If $A$ is countably infinite, then we can list the elements in $A$, }أ��-W�������"�C��� �00�!�U嚄l�'}�� -F�NQ gvPC���S�:|����ա���ʛ������v���|�Z���uo�2�a`ynް�K��gS��v�������*��P~Ē�����&63 If $A$ has only a finite number of elements, its cardinality is simply the If $A_1, A_2,\cdots$ is a list of countable sets, then the set $\bigcup_{i} A_i=A_1 \cup A_2 \cup A_3\cdots$ It turns out we need to distinguish between two types of infinite sets, ����RJ�IR�� $$B = \{b_1, b_2, b_3, \cdots \}.$$ is also countable. refer to Figure 1.16 in Problem 2 to see this pictorially). Sets such as $\mathbb{N}$ and $\mathbb{Z}$ are called countable, thus $B$ is countable. Any set containing an interval on the real line such as $[a,b], (a,b], [a,b),$ or $(a,b)$, Definition13.1settlestheissue. Theorem13.1 Thereexistsabijection f :N!Z.Therefore jNj˘jZ. Thus, we have. /Length 1868 Assume $B$ is uncountable. should also be countable, so a subset of a countable set should be countable as well. $$\mathbb{Q}=\bigcup_{i \in \mathbb{Z}} \bigcup_{j \in \mathbb{N}} \{ \frac{i}{j} \}.$$. The above theorems confirm that sets such as $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$ and their If you are less interested in proofs, you may decide to skip them. Becausethebijection f :N!Z matches up Nwith Z,itfollowsthat jj˘j.Wesummarizethiswithatheorem. set which is a contradiction. This is because we can write However, as we mentioned, intervals in $\mathbb{R}$ are uncountable. is concerned, this guideline should be sufficient for most cases. Before discussing We can extend the same idea to three or more sets. endobj And n (A) = 7. Some sets that are not countable include ℝ, the set of real numbers between 0 and 1, and ℂ. If set A is countably infinite, then | A | = | N |. << In particular, one type is called countable, On the other hand, you cannot list the elements in $\mathbb{R}$, Thus by applying $$A = \{a_1, a_2, a_3, \cdots \},$$ 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!). One may be tempted to say, in analogy with finite sets, that all ?�iOQ���i�ʫ�.��_���׉!ә�. For example, if $A=\{2,4,6,8,10\}$, then $|A|=5$. $$|W|=10$$ -�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%�>'���. All of the sets have the same cardinality as the natural numbers ℕ. $$\>\>\>\>\>\>\>+\sum_{i < j < k}\left|A_i\cap A_j\cap A_k\right|-\ \cdots\ + \left(-1\right)^{n+1} \left|A_1\cap\cdots\cap A_n\right|.$$, $= |W| + |R| + |B|- |W \cap R| - |W \cap B| - |R \cap B| + |W \cap R \cap B|$. $$|A \cup B |=|A|+|B|-|A \cap B|.$$ JHY!Ǭ4�5�8s�-�����D��~�)�X�`ŐN/�*��u��yߺ��6�ԏ@�Z����R�9�pf�C�`Te�n�K�T��`X*��� ��T� F,���˙�� �M�zϞ��=Fr������JLL`�W��EO��& $$|R|=8$$ S*~����7ׇ�E��bba&�Eo�oRB@3a͜9dQ�)ݶ�PSa�a�u��,�nP{|���Jq(jS�z1?m��h�^�aG?c��3>������1p+!��$�R��`V�:�$��� �x�����2���/�d 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. ����{i�V�_�����A|%�v��{&F �B��oA�)QC|*i�P@c���$[B��X>�ʏ)+aK6���� -o��� �6� ;�I-#�a�F�*<9���*]����»n�s鿻摞���H���q��ѽ��n�WB_�S����c�ju�A:#�N���/u�,�0ki��2��:����!W�K/��H��'��Ym�R2n�)���2��;Û��&����:��'(��yt�Jzu�*Ĵ�1�&1}�yW7Q���m�M(���Q Ed ���ˀ������C�s� Ӌ��&�Qh��Ou���cJ����>���I6�'�/m��o��m�?R�"o�ͽP�����=�N��֩���&�5��y&���0 �$�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� then talk about infinite sets. >> thus $B$ is countable. The function f : … SetswithEqualCardinalities 219 N because Z has all the negative integers as well as the positive ones. $$|W \cup B \cup R|=21.$$ Since each $A_i$ is countable we can

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