2) The set of all whole numbers less than 20. 1) The set of all positive even numbers. Instead, we use the more appropriate set-builder notation which describes what elements are contained in the set. A set is countable if and only if it is finite or countably infinite. 4) The set of all odd natural numbers less than 15. \(d\) is the created number which will never be on the list. Now I need to come up with a function to accomplish this mapping to the negative integers, and after some thinking, I come up with \(f(n)=-\frac{n+1}{2}.\) We first discuss cardinality for finite sets and then talk about infinite sets. Legal. Teachoo is free. A set \(A\) is countably infinite if and only if set \(A\) has the same cardinality as \(\mathbb{N}\) (the natural numbers). \(\mathbb{Z} \mbox{  and } \mathbb{Q} \) are countably infinite sets. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Infinite set; A set which contains unlimited number of elements is called an infinite set. 5) The set of all letters in the word ‘computer’. The continuum hypothesis is the statement that there is no set whose cardinality is strictly between that of \(\mathbb{N} \mbox{   and   } \mathbb{R}\). The proof that a set cannot be mapped onto its power set is similar to the Russell paradox, named for Bertrand Russell. Write this (infinite) list, and as it's written, we will create a number that is NOT on that list. Let R = {whole numbers between 5 and 45} Have questions or comments? So, for the second number on the list, we see the second digit is a 5, and we choose a 0 for the second digit of our number being created. N is an infinite set and is the same as Z +. A is the set of fractions. The operations of basic set theory can be used to produce more examples of uncountably infinite sets: If A is a subset of B and A is uncountable, then so is B. Examples: T = {x : x is a triangle} N is the set of natural numbers. Which of the following sets are finite or infinite ? There is a nice proof you may have seen where all the fractions are listed in an endless matrix and it can be seen that a path can be drawn to cover all the fractions. (This is an example, not a proof. Example: {10, 20, 30, 40} has an order of 4. B = {1, 2, 3, 4, 5, 6, 7, 8, 9, ………. } 5) The set of all letters in the word ‘computer’. Watch the recordings here on Youtube! It can be shown that this function is well-defined and a bijection.). On signing up you are confirming that you have read and agree to Learn Science with Notes and NCERT Solutions, Number of elements in set - 2 sets (Direct), Number of elements in set - 2 sets - (Using properties), Proof - where properties of sets cant be applied,using element. \[ \mathbb{N}=\{1,2,3,4,...\}\mbox{      is the set of Natural Numbers, also known as the Counting Numbers}.\], \(\mathbb{N}\) is an infinite set and is the same as \( \mathbb{Z}^+.\). 5th number:   0.777888222.....           our number that we are creating 0.00110 Also \(|\emptyset|=0.\). Furthermore, we designate the cardinality of countably infinite sets as  \(\aleph_0\)  ("aleph null"). is said to be an Infinite Set. For small finite sets, we can often describe the set by writing the elements within curly braces separated by … Any subset of a countable set is countable. Example: B = Set of all natural numbers. Examples of finite set: 1. The set of integers \(\mathbb{Z}\) and its subset, set of even integers \(E = \{\ldots -4, -2, 0, 2, 4, \ldots\}.\), The function \(f: \mathbb{Z} \to E\) given by \(f(n) = 2 n\) is one-to-one and onto. etc. Let P = {5, 10, 15, 20, 25, 30} Then, P is a finite set and n(P) = 6. For example, consider a set Infinity occurs in the real world as the possibility to continue forever, that is, potential infinity. So, for the third number on the list, we see the third digit is a 0, and we choose a 1 for the third digit of our number being created. This provides a more straightforward proof that the entire set of real numbers is uncountable. Learn about Sets on our Youtube Channel - https://you.tube/Chapter-1-Class-11-Sets. 3) The set of all positive integers which are multiples of 3. If \(S\) is countably infinite and \(A \subseteq S\) then \(A\) is countable. 2. Login to view more pages. Thinking of how to match the natural numbers to the integers, I see how the even natural numbers could be used for the positive integers, like this: Teachoo provides the best content available! \(P=\{\mbox{olives, mushrooms, broccoli, tomatoes}\}\) and \(Q=\{\mbox{Jack, Queen, King, Ace}\}.\). The cardinality of a set is denoted by $|A|$. Let \(n_i\) be the \(i\)th smallest index such that \(x_{n_i} \in A\). The cardinality of an infinite set is n (A) = … An infinite set is a set which is not finite. \(\aleph_1=|\mathbb{R}|=|(0,1)|= |\scr{P}(\mathbb{N})|\)                 cardinality of the "lowest" uncountably infinite sets; also known as "cardinality of the continuum". If the elements of a set cannot be counted, i.e. It is not possible to explicitly list out all the elements of an infinite set. Then create \(d_n= \begin{cases}  1 & \text{if } a_{nn} \neq 0\ \\ 0 & \text{if } a_{nn}=0\end{cases}\) Transfinite numbers are used to describe the cardinalities of "higher & higher" infinities. (We choose a 0 unless the digit we are comparing to is a 0 and then we choose a 1.) For finite sets the order (or cardinality) is the number of elements . 2nd number:  0.051023237.....           our number that we are creating 0.00 \[\mbox{For all sets }A,B,C, \qquad \qquad \mbox{ if  }|A|=|B| \mbox{  and  } |B|=|C| \mbox{  then  } |A|=|C|\]. An infinite set that can be put into a one-to-one correspondence with \(\mathbb{N}\) is countably infinite. Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. Cardinality is transitive (even for infinite sets). Finite sets and countably infinite are called countable.