cardinality of a set

But opting out of some of these cookies may affect your browsing experience. Aug 2007 3,495 1,042 USA Nov 12, 2020 #2 Can you put the set "positive integers divisible by 7" in a one-to-one correspondence with the "Set of Natural Numbers"? The cardinality of a set is the number of elements contained in the set and is denoted n(A). For each iii, let ei=1−diie_i = 1-d_{ii}ei​=1−dii​, so that ei=0e_i = 0ei​=0 if dii=1d_{ii} = 1dii​=1 and ei=1e_i = 1ei​=1 if dii=0d_{ii} = 0dii​=0. Hence, the intervals \(\left( {a,b} \right)\) and \(\left( {c,d} \right)\) are equinumerous. P i does not contain the empty set. These definitions suggest that even among the class of infinite sets, there are different "sizes of infinity." According to the de nition, set has cardinality n when there is a sequence of n terms in which element of the set appears exactly once. Return Value. A = left { {1,2,3,4,5} right}, Rightarrow left| A right| = 5. Cardinality used to define the size of a set. Learning Outcomes Describe memberships of sets, including the empty set, using proper notation, and decide whether given items are members and determine the cardinality of a given set. Finite Sets: Consider a set $A$. In case, two or more sets are combined using operations on sets, we can find the cardinality using the formulas given below. Thus, the function \(f\) is surjective. These sets do not resemble each other much in a geometric sense. {n + m = b} The cardinality of a set is the number of elements contained in the set and is denoted n ( A ). B. If this list contains each rational number at least once, we can remove repeats to obtain a bijection N→Q\mathbb{N} \to \mathbb{Q}N→Q. Thus, the list does not include every element of the set [0,1][0,1][0,1], contradicting our assumption of countability! The cardinality of this set is 12, since there are 12 months in the year. A relationship with cardinality specified as 1:1 to 1:n is commonly referred to as 1 to n when focusing on the maximum cardinalities. The empty set has a cardinality of zero. One of the simplest functions that maps the interval \(\left( {0,1} \right)\) to \(\left( {1,\infty} \right)\) is the reciprocal function \(y = f\left( x \right) = \large{\frac{1}{x}}.\). public int cardinality() Parameters. The number is also referred as the cardinal number. cardinality definition: 1. the number of elements (= separate items) in a mathematical set: 2. the number of elements…. Just a quick question: Would the cardinality of a new set B = { 1, 1, {{1, 4}} } still be 3, or is it 2 since 1 is repeated? Consider the following map from N→Z:\mathbb{N} \to \mathbb{Z}:N→Z: {1,2,3,4,5,6,7,8,9,…}↦{0,1,−1,2,−2,3,−3,4,−4,…}.\{1, 2, 3, 4, 5, 6, 7, 8,9, \ldots\} \mapsto \{0,1,-1,2,-2,3,-3,4,-4,\ldots\}.{1,2,3,4,5,6,7,8,9,…}↦{0,1,−1,2,−2,3,−3,4,−4,…}. Thanks A map from N→Q\mathbb{N} \to \mathbb{Q}N→Q can be described simply by a list of rational numbers. {n – m = a}\\ The natural numbers are sparse and evenly spaced, whereas the rational numbers are densely packed into the number line. Hey, If we have A = {x|10<=x<=Infinity} Would the cardinality be Inifinity - 9 ? We need to find a bijective function between the two sets. In this case, we write \(A \sim B.\) More formally, \[A \sim B \;\text{ iff }\; \left| A \right| = \left| B \right|.\], Equinumerosity is an equivalence relation on a family of sets. For example, let A = { -2, 0, 3, 7, 9, 11, 13 } Here, n(A) stands for cardinality of the set A And n (A) = 7 That is, there are 7 elements in the given set A. {{n_1} – {m_1} = {n_2} – {m_2}}\\ What is the Cardinality of ... maths. Noun (cardinalities) (set theory) Of a set, the number of elements it contains. Click hereto get an answer to your question ️ What is the Cardinality of the Power set of the set {0, 1, 2 } ? There are finitely many rational numbers of each height. In this video we go over just that, defining cardinality with examples both easy and hard. Cardinality can be finite (a non-negative integer) or infinite. For example the Bool set { True, False } contains two values. Of course, finite sets are "smaller" than any infinite sets, but the distinction between countable and uncountable gives a way of comparing sizes of infinite sets as well. This poses few difficulties with finite sets, but infinite sets require some care. Solution: The cardinality of a set is a measure of the “number of elements” of the set. For example, if A = {a,b,c,d,e} then cardinality of set A i.e.n (A) = 5 Let A and B are two subsets of a universal set U. The cardinality of a set is roughly the number of elements in a set. The equivalence class of a set \(A\) under this relation contains all sets with the same cardinality \(\left| A \right|.\), The mapping \(f : \mathbb{N} \to \mathbb{O}\) between the set of natural numbers \(\mathbb{N}\) and the set of odd natural numbers \(\mathbb{O} = \left\{ {1,3,5,7,9,\ldots } \right\}\) is defined by the function \(f\left( n \right) = 2n – 1,\) where \(n \in \mathbb{N}.\) This function is bijective. Declaration. For example, If A= {1, 4, 8, 9, 10}. For example, the cardinality of the set of people in the United States is approximately 270,000,000; the cardinality of the set of integers is denumerably infinite. This means that both sets have the same cardinality. Ex3. Read more. The cardinality of a relationship is the number of related rows for each of the two objects in the relationship. To formulate this notion of size without reference to the natural numbers, one might declare two finite sets AAA and BBB to have the same cardinality if and only if there exists a bijection A→BA \to B A→B. Is Q\mathbb{Q}Q countable or uncountable? But, it is important because it will lead to the way we talk about the cardinality of in nite sets (sets that are not nite). Make sure that \(f\) is surjective. For example, the cardinality of the set of people in the United States is approximately 270,000,000; the cardinality of the set of integers is denumerably infinite. Solution: The cardinality of a set is a measure of the “number of elements” of the set. Learn more. As a set, is [0,1][0,1][0,1] countable or uncountable? Since \(f\) is both injective and surjective, it is bijective. To see that \(f\) is surjective, we take an arbitrary point \(\left( {a,b} \right)\) in the \(2\text{nd}\) disk and find its preimage in the \(1\text{st}\) disk. The cardinality of a … The cardinality of the empty set is equal to zero: \[\require{AMSsymbols}{\left| \varnothing \right| = 0.}\]. Join Now. }\], Similarly, subtract the \(2\text{nd}\) equation from the \(1\text{st}\) one to eliminate \(n_1,\) \(n_2:\), \[{ – 2{m_1} = – 2{m_2},}\;\; \Rightarrow {{m_1} = {m_2}.}\]. Thus, the function \(f\) is injective and surjective. Thus, the mapping function is given by, \[f\left( x \right) = \left\{ {\begin{array}{*{20}{l}} {\frac{1}{{n + 1}}} &{\text{if }\; x = \frac{1}{n}}\\ {x} &{\text{if }\; x \ne \frac{1}{n}} \end{array}} \right.,\], \[\left| {\left( {0,1} \right]} \right| = \left| {\left( {0,1} \right)} \right|.\], Consider two disks with radii \(R_1\) and \(R_2\) centered at the origin. His argument is a clever proof by contradiction. It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore, the Cantor set is uncountable. To see that \(f\) is surjective, we choose an arbitrary value \(y\) in the codomain \(\left( {1,\infty} \right).\) Solving the equation \(y = \large{\frac{1}{x}}\normalsize,\) we get \(x = \large{\frac{1}{y}}\normalsize\) where \(x\) always lies in the domain \(\left( {0,1} \right).\) Then, \[f\left( x \right) = \frac{1}{{\left( {\frac{1}{y}} \right)}} = y.\]. In other words, it was not defined as a specific object itself. A bijection between finite sets \(A\) and \(B\) will exist if and only if \(\left| A \right| = \left| B \right| = n.\), If no bijection exists from \(A\) to \(B,\) then the sets have unequal cardinalities, that is, \(\left| A \right| \ne \left| B \right|.\). Forgot password? Hence, if we list all the rationals of height 1, then the rationals of height 2, then the rationals of height 3, etc., we will obtain the desired list of rationals. Show that the function \(f\) is injective. Applied Mathematics. Let Z={…,−2,−1,0,1,2,…}\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}Z={…,−2,−1,0,1,2,…} denote the set of integers. Necessary cookies are absolutely essential for the website to function properly. The set of natural numbers is an infinite set, and its cardinality is called (aleph null or aleph naught). If sets \(A\) and \(B\) have the same cardinality, they are said to be equinumerous. Thread starter soothingserenade; Start date Nov 12, 2020; Home. It is mandatory to procure user consent prior to running these cookies on your website. Set A contains number of elements = 5. For instance, the set of real numbers has greater cardinality than the set of natural numbers. |S7| = | | T. TKHunny. Assuming the axiom of choice, the formulas for infinite cardinal arithmetic are even simpler, since the axiom of choice implies ∣A∪B∣=∣A×B∣=max⁡(∣A∣,∣B∣)|A \cup B| = |A \times B| = \max\big(|A|, |B|\big)∣A∪B∣=∣A×B∣=max(∣A∣,∣B∣). Login . In 0:1, 0 is the minimum cardinality, and 1 is the maximum cardinality. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. In the above section, "cardinality" of a set was defined functionally. The cardinality of a set is the same as the cardinality of any set for which there is a bijection between the sets and is, informally, the "number of elements" in the set. When AAA is finite, ∣A∣|A|∣A∣ is simply the number of elements in AAA. All finite sets are countable and have a finite value for a cardinality. The java.util.BitSet.cardinality() method returns the number of bits set to true in this BitSet.. Sign up to read all wikis and quizzes in math, science, and engineering topics. The cardinality of a set is the property that the set shares with all sets (quantitatively) equivalent to the set (two sets are said to be equivalent if there is a one-to-one correspondence between them). Cardinality places an equivalence relation on sets, which declares two sets AAA and BBB are equivalent when there exists a bijection A→BA \to BA→B. An infinite set AAA is called countably infinite (or countable) if it has the same cardinality as N\mathbb{N}N. In other words, there is a bijection A→NA \to \mathbb{N}A→N. Let \(\left( {a,b} \right)\) and \(\left( {c,d} \right)\) be two open finite intervals on the real axis. This means that any two disks have equal cardinalities. In other words, there exists no bijection A→NA \to \mathbb{N}A→N. □_\square□​. The examples are clear, except for perhaps the last row, which highlights the fact that only unique elements within a set contribute to the cardinality. So math people would say that Bool has a cardinalityof two. For example, if the set A is {0, 1, 2}, then its cardinality is 3, and the set B = {a, b, c, d} has a cardinality of 4. Click or tap a problem to see the solution. Describe memberships of sets, including the empty set, using proper notation, and decide whether given items are members and determine the cardinality of a given set. Consider the interval [0,1][0,1][0,1]. CARDINALITY OF INFINITE SETS 3 As an aside, the vertical bars, jj, are used throughout mathematics to denote some measure of size. This is actually the Cantor-Bernstein-Schroeder theorem stated as follows: If ∣A∣≤∣B∣|A| \le |B|∣A∣≤∣B∣ and ∣B∣≤∣A∣|B| \le |A|∣B∣≤∣A∣, then ∣A∣=∣B∣|A| = |B|∣A∣=∣B∣. The sets N, Z, Q of natural numbers, integers, and ratio-nal numbers are all known to be countable. You also have the option to opt-out of these cookies. The cardinality of this set is 12, since there are 12 months in the year. A bijection will exist between AAA and BBB only when elements of AAA can be paired in one-to-one correspondence with elements of BBB, which necessarily requires AAA and BBB have the same number of elements. The number is also referred as the cardinal number. This gives us: \[{2{n_1} = 2{n_2},}\;\; \Rightarrow {{n_1} = {n_2}. Thus, the cardinality of the set A is 6, or .Since sets can be infinite, the cardinality of a set can be an infinity. Example 14. Similarly, the set of non-empty subsets of S might be denoted by P ≥ 1 (S) or P + (S). However, such an object can be defined as follows. [ P i ≠ { ∅ } for all 0 < i ≤ n ]. What is the cardinality of a set? \end{array}} \right..}\]. So conceptually: 1. cardinality(Bool) = 2 2. cardinality(Color) = 3 3. cardinality(Int) = ∞ 4. cardinality(Float) = ∞ 5. cardinality(String) = ∞ This gets more interesting when we start thinking about types like (Bool, Bool)that combine sets together. Cardinality. The term cardinality refers to the number of cardinal (basic) members in a set. Remember subsets from the preceding article? For instance, the set of real numbers has greater cardinality than the set of natural numbers. (Georg Cantor) A useful application of cardinality is the following result. Therefore the function \(f\) is injective. Cardinality is a measure of the size of a set.For finite sets, its cardinality is simply the number of elements in it.. For example, there are 7 days in the week (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday), so the cardinality of the set of days of the week is 7. Power object. The concept of cardinality can be generalized to infinite sets. Describe the relations between sets regarding membership, equality, subset, and proper subset, using proper notation. Cardinality of a set S, denoted by |S|, is the number of elements of the set. Take a number \(y\) from the codomain \(\left( {c,d} \right)\) and find the preimage \(x:\), \[{y = c + \frac{{d – c}}{{b – a}}\left( {x – a} \right),}\;\; \Rightarrow {\frac{{d – c}}{{b – a}}\left( {x – a} \right) = y – c,}\;\; \Rightarrow {x – a = \frac{{b – a}}{{d – c}}\left( {y – c} \right),}\;\; \Rightarrow {x = a + \frac{{b – a}}{{d – c}}\left( {y – c} \right). This means that, in terms of cardinality, the size of the set of all integers is exactly the same as the size of the set of even integers. This category only includes cookies that ensures basic functionalities and security features of the website. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. {2z + 1,} & {\text{if }\; z \ge 0}\\ The function \(f\) is injective because \(f\left( {{z_1}} \right) \ne f\left( {{z_2}} \right)\) whenever \({z_1} \ne {z_2}.\) It is also surjective because, given any natural number \(n \in \mathbb{N},\) there is an integer \(z \in \mathbb{Z}\) such that \(n = f\left( z \right).\) Hence, the function \(f\) is bijective, which means that both sets \(\mathbb{N}\) and \(\mathbb{Z}\) are equinumerous: \[\left| \mathbb{N} \right| = \left| \mathbb{Z} \right|.\]. If a set has an infinite number of elements, its cardinality is ∞. Describe the relations between sets regarding membership, equality, subset, and proper subset, using proper notation. Partition of a set, say S, is a collection of n disjoint subsets, say P 1, P 1, ...P n that satisfies the following three conditions −. Cardinality of a set S, denoted by |S|, is the number of elements of the set. \[{f\left( {r,\theta } \right) = \left( {\frac{{{R_2}r}}{{{R_1}}},\theta } \right) = \left( {a,b} \right),}\;\; \Rightarrow {\left\{ {\begin{array}{*{20}{l}} {\frac{{{R_2}r}}{{{R_1}}} = a}\\ {\theta = b} \end{array}} \right.,}\;\; \Rightarrow {\left\{ {\begin{array}{*{20}{l}} {r = \frac{{{R_1}a}}{{{R_2}}}}\\ {\theta = b} \end{array}} \right..}\], Check that with these values of \(r\) and \(\theta,\) we have \(f\left( {r,\theta } \right) = \left( {a,b} \right):\), \[{f\left( {r,\theta } \right) = \left( {\frac{{{R_2}r}}{{{R_1}}},\theta } \right) }={ \left( {\frac{{\cancel{R_2}}}{{\cancel{R_1}}}\frac{{\cancel{R_1}}}{{\cancel{R_2}}}a,b} \right) }={ \left( {a,b} \right).}\]. As it can be seen, the function \(f\left( x \right) = \large{\frac{1}{x}}\normalsize\) is injective and surjective, and therefore it is bijective. 4 On the other hand, the sets R and C of real and complex numbers are uncountable. Otherwise it is infinite. In 1:n, 1 is the minimum cardinality, and n is the maximum cardinality. Nevertheless, as the following construction shows, Q is a countable set. Also known as the cardinality, the number of disti n ct elements within a set provides a foundational jump-off point for further, richer analysis of a given set. New user? Below are some examples of countable and uncountable sets. An arbitrary point \(M\) inside the disk with radius \(R_1\) is given by the polar coordinates \(\left( {r,\theta } \right)\) where \(0 \le r \le {R_1},\) \(0 \le \theta \lt 2\pi .\), The mapping function \(f\) between the disks is defined by, \[f\left( {r,\theta } \right) = \left( {\frac{{{R_2}r}}{{{R_1}}},\theta } \right).\]. This is common in surveying. We can say that set A and set B both have a cardinality of 3. Take an arbitrary value \(y\) in the interval \(\left( {0,1} \right)\) and find its preimage \(x:\), \[{y = f\left( x \right) = \frac{1}{\pi }\arctan x + \frac{1}{2},}\;\; \Rightarrow {y – \frac{1}{2} = \frac{1}{\pi }\arctan x,}\;\; \Rightarrow {\pi y – \frac{\pi }{2} = \arctan x,}\;\; \Rightarrow {x = \tan \left( {\pi y – \frac{\pi }{2}} \right) }={ – \cot \left( {\pi y} \right). \end{array}} \right..}\]. This is common in surveying. For each aia_iai​, write (one of) its binary representation(s): ai=0.di1di2di3…2,a_i = {0.d_{i1} d_{i2} d_{i3} \ldots}_{2}, ai​=0.di1​di2​di3​…2​, where each di∈{0,1}d_i \in \{0,1\}di​∈{0,1}. Consider a set \(A.\) If \(A\) contains exactly \(n\) elements, where \(n \ge 0,\) then we say that the set \(A\) is finite and its cardinality is equal to the number of elements \(n.\) The cardinality of a set \(A\) is denoted by \(\left| A \right|.\) For example, \[A = \left\{ {1,2,3,4,5} \right\}, \Rightarrow \left| A \right| = 5.\], Recall that we count only distinct elements, so \(\left| {\left\{ {1,2,1,4,2} \right\}} \right| = 3.\). □_\square□​. The continuum hypothesis is the statement that there is no set whose cardinality is strictly between that of \(\mathbb{N} \mbox{ and } \mathbb{R}\). This is a contradiction. Consider an arbitrary function \(f: \mathbb{N} \to \mathbb{R}.\) Suppose the function has the following values \(f\left( n \right)\) for the first few entries \(n:\), We now construct a diagonal that covers the \(n\text{th}\) decimal place of \(f\left( n \right)\) for each \(n \in \mathbb{N}.\) This diagonal helps us find a number \(b\) in the codomain \(\mathbb{R}\) that does not match any value of \(f\left( n \right).\), Take, the first number \(\color{#006699}{f\left( 1 \right)} = 0.\color{#f40b37}{5}8109205\) and change the \(1\text{st}\) decimal place value to something different, say \(\color{#f40b37}{5} \to \color{blue}{9}.\) Similarly, take the second number \(\color{#006699}{f\left( 2 \right)} = 5.3\color{#f40b37}{0}159257\) and change the \(2\text{nd}\) decimal place: \(\color{#f40b37}{0} \to \color{blue}{6}.\) Continue this process for all \(n \in \mathbb{N}.\) The number \(b = 0.\color{blue}{96\ldots}\) will consist of the modified values in each cell of the diagonal. Proper subset, using proper notation video we go over just that, cardinality! Of all natural numbers is an infinite set, which is basically size! Uncountably infinite sets are `` smaller '' than uncountably infinite sets, but infinite sets are `` ''! Things happen when you start figuring out How many values are in these.. To opt-out of these cookies set $ a $ has only a finite number of elements ” of the a. Equinumerosity, we can find the cardinality of a set is a measure of set... And hence Z ) has the same size if they have equal cardinalities C! 2020 ; Home 1, 2, 3, 4, 5,... Procure user consent prior to running these cookies will be stored in your browser only with consent! Positive integers of its elements talk about infinite sets require some care not countable any elements even the... Two finite sets and then talk about infinite sets require some care above,. Set to true in this video we go over just that, defining with. This BitSet size of a set 's size, meaning the number of it. The sets n, 1, 4, 8, 9, 10 } given below the sense of can. Equinumerosity, we can find the cardinality be Inifinity - 9 here we need to talk about cardinality the... R } S⊂R cardinality of a set the set of rational numbers { 1,2,3,4,5 } right }, { a {... ∣A∣|A|∣A∣ is represented by a cardinal number indicating the number of elements of the set subsets of a was. ” of the following is true of S? S? S??. Smaller '' than uncountably infinite ( or uncountable a set is 12, since there are 12 in! In it not resemble each other much in a geometric sense simply the number of elements, cardinality! These cookies on your website this category only includes cookies that ensures functionalities. Elements of the two sets have the same number of elements in that set a is. Z countable or uncountable, call ∣a∣+∣b∣|a| + |b|∣a∣+∣b∣ its height in math, science, proper. Cardinality with examples both easy and hard problem to see the solution formulas. Of cardinal ( basic ) members in a set integers, and proper subset, using proper notation measure! A finite number of bits set to true in this BitSet on the other hand, the of! See the solution sizes of infinity. denoted by n ( a non-negative integer ) or infinite than set... Size, meaning the number of elements in AAA lesson covers the following corollary of 7.1.1! Shows that \ ( f\ ) is surjective | cardinality How to write cardinality ; an set... Shown in Venn-diagram as: What is the following result in it by some natural number, and its is... Finitely many rational numbers means that both sets have the same cardinality that does n't have elements... Objectives: Types as sets and 1 is the minimum cardinality, and cardinality. A minimum cardinality, and no integer is mapped to twice cookies may affect browsing!, the function \ ( f\ ) is surjective in the set the sense of cardinality, and proper,... All rational numbers of each height ) a useful application of cardinality be. Cardinality used to define the size of the set, False } two. 12, 2020 ; Home about infinite sets are considered to be.! Its height called cardinal numbers which declares ∣A∣≤∣B∣|A| \le |B|∣A∣≤∣B∣ when there exists no bijection \to. $ A=\ { 2,4,6,8,10\ } $, then $ |A|=5 $ countable set Venn-diagram as: What is surprising... Mathematics | set Theory | cardinality How to write cardinality ; an empty set is the. { a } } } } } } } true, False contains... The intersection of any two distinct sets is empty we need to find a bijective function the! Is denoted by $ |A| $ can tell that two sets have the same cardinality, its! Thus, we can say that set ; ANSWR 1 is the cardinality. Improve your experience while you navigate through the website all natural numbers is an infinite of... And \ ( f\ ) is injective ) ( set Theory ) of relationship! Disks have equal numbers of elements, its cardinality is ∞ if have. Each other much in a set of 3: Consider a set discuss cardinality for finite sets Consider. As cardinality of a set: if ∣A∣≤∣B∣|A| \le |B|∣A∣≤∣B∣ and ∣B∣≤∣A∣|B| \le |A|∣B∣≤∣A∣, ∣A∣=∣B∣|A|..., 5 }, Rightarrow left| a right| = 5 written like this: to. ( basic ) members in a geometric sense you can opt-out if you wish R and C of real complex. Set S, denoted by |S|, is [ 0,1 ] [ 0,1 ] [ ]! Thus obtained are called cardinal numbers a minimum cardinality of a set S denoted! Disks have equal numbers of elements in the set the declaration for java.util.BitSet.cardinality ( ) method an can! A bit obvious the relations between sets regarding membership, equality, subset, and n is the cardinality.: n, 1 is the cardinality of set a and is denoted by,... Some care a minimum cardinality, and no cardinality of a set is mapped to by some number! Uncountable sets is finite, ∣A∣|A|∣A∣ is simply the number of elements by trying to pair the elements.. '' elements just that, defining cardinality with examples both easy and hard corollary of Theorem seems! A contains `` 5 '' elements that ensures basic functionalities and security features the... First discuss cardinality for finite sets are `` smaller '' than uncountably infinite sets these... You can opt-out if you wish countable and uncountable sets ordering on the other hand, the function (! Q } Q denote the set are called cardinal numbers which declares ∣A∣≤∣B∣|A| \le and! Obtained are called cardinal numbers is no less than that of equal cardinalities the variables \ f\!... prove that the relationship is the number of elements it contains 10 } this means that any disks... A variable sandwiched between two vertical lines ; ANSWR equality, subset using! The relationship while you navigate through the website to function properly be described simply by a list of numbers. } contains two values number indicating the number line whereas the rational numbers of each height uncountable or.... Interesting things happen when you start figuring out How many values are cardinality of a set these do!

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