How To Find Cardinal Number In A Set
Fundamental Number: Definition, Diagram, Types, Examples
Fundamental Numbers: Central numbers are used for counting dissimilar things. These are also known as cardinals. These are the natural numbers that start from \(ane\) and keep sequentially and are not fractions. The word cardinals refer to 'how many of anything is existing in a group. Similar if nosotros want to count the number of apples present in the basket, we must make utilise of these numbers, such as \(1, ii, 3, 4, 5,\)…. and and then on. Numbers assist to count the number of things; people present in the place or a group.
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Set
Definition: A gear up is a collection of well-defined objects.
Here, the word 'well defined' means it should be possible to make up one's mind whether an object does or does non belong to a specific collection.
The objects of the fix are known as its elements or members. The symbol \(\in \) is used to hateful "is an chemical element of", and the symbol \( \notin \) is used to denote "is not an element of".
Example: In set \(A = \left\{={a,~b,~c} \right\};\) \(a, b\) and \(c\) are the elements of set \(A\).
Nosotros write, \(a \in A\), for \("a\) is an element of \(A"\).
\(b \in A\), for \("b\) is an element of \(A"\).
\(c \in A\), for \("c\) is an element of \(A"\).
And \(d \in A\), for \("d\) is not an chemical element of \(A"\).
Ways to Stand for a Set
To represent a set up, we utilize the post-obit methods:
Description Method
In this method, we write the (well-defined) description of the elements of the set and this clarification is enclosed in curly brackets.
Example:
1. The 'set of prime number numbers less than \(ten'\) is written as {prime numbers less than \(10\)}.
Note that \(0 \in \) {whole numbers less than \(ten\)} while \(ten \notin \) {whole numbers less than \(ten\)}.
2. The set of even integers is written as {fifty-fifty integers}.
Notation that \(8 \in \) {fifty-fifty integers} while \(5 \notin \) {fifty-fifty integers}.
3. The set of colours of the rainbow is written as {colours of rainbow}.
Note that blue \( \in \) {colour of rainbow} while blackness \( \notin \) {colours of rainbow}.
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Roster Method
In this method, we list all the elements of the gear up and separate them by commas. The pair of flowers (curly) brackets enclose the listing.
Case: The ready \(A\) of even whole numbers less than \(x\) in the roster method is written as:
\(A = \left\{ {0,~2,~4,~6,~viii} \right\}\).
Note that \(0,\,6 \in A\) while \(10 \notin A\).
\(A = \left\{ {a,~e,~i,~o,~u} \right\}\) represents the set of vowels.
Set Builder Class
In prepare-builder form, we write a variable (say \(x\)) representing any number of the set followed by a property satisfied past each element of the grouping and enclose it in flower (curly) brackets.
If \(A\) is the set consisting of elements \(ten\) having property \(p\), nosotros write:
\(A = \{ 10|ten\) has property \(p\} \)
Which is read as 'the ready of elements \(x\) such that \(x\) has property \(p'\).
The symbol \('|\)' stands for the words 'such that'. Sometimes, we use the logo \(':'\) in place of the symbol \('|'\).
Example: The set \(A\) of whole numbers less than \(10\) in the fix-builder form is written every bit
\(A = \{ x|ten \in W,~x < 10\} \)
Information technology is read every bit 'the set of elements \(x\) such that \(x\) belong to \(Due west\) and \(x\) is less than \(10'\).
Venn Diagram
The relationship betwixt the sets can be represented by means of diagrams which are known as Venn Diagrams.
The Venn Diagram consists of rectangles and airtight curves, commonly circles. The universal set is represented commonly past a rectangle and the subsets of it by circles.
In Venn Diagrams, the elements of a set are commonly written in their corresponding circles.
Case: \(U = \left\{ {one,~2,~3,~ \ldots .,~10} \right\}\) is the universal set of which \(A = \left\{ {~2,~four,~half-dozen,~8,~10} \right\}\) is a subset.
Know About Ordinal Numbers
Cardinal Number
Definition: The symbol to represent the number of elements in a fix \(A\) is \(due north (A)\). it is known as a primal number of set \(A\).
A set may have
1. No elements
2. One or more elements with
a. A finite number of elements.
b. An infinite number of elements.
Cardinal Numbers for Different Sets
According to the number of elements independent in a fix, we define dissimilar types of sets as follows:
Finite Set
A set that contains a express (counted) number of different elements is called a finite set. In other words, a set is called a finite fix if the counting of its various parts comes to an end.
Case: \(A = \left\{ {~a,~b,~c,~d,~e} \right\}\) is a finite set having five elements. The cardinal number of set \(A\), denoted by \(n(A)\) is \(5\).
\(B = \{ x|x \in West,~ten < vii\} \) is a finite set having seven elements. The primal number of gear up \(B\), denoted by \(n(B)\) is \(vii\).
\(C = \left\{ {1,~three,~5,~ \ldots .,~10} \correct\}\) is a finite set. The primal number of set \(C\), denoted past \(n(C)\) is \(101\).
Practice Exam Questions
Infinite Set
A set that contains an unlimited (uncountable) number of different elements is chosen an infinite set. In other words, a fix is called if the counting of its diverse elements does non come to an end.
Example: the set up of natural numbers \(N = \left\{ {~1,~2,~3,~iv,~…..} \correct\}\)
The set whole numbers \(W = \left\{ {~0,~i,~2,~3,~…..} \right\}\)
The set of integers \(I = \left\{ {~-3,~-two,~-1,~0,~1,~2,~iii,~……} \right\}\)
Each of the above sets is an space prepare. Hence, the cardinal number for each of them is an infinite number.
Empty Set
The set containing no elements is a blank or void, or null fix. It is denoted past the symbol \(\phi \) or \(\left\{ ~ \right\}\).
Example: The gear up of cats with two tails is the empty set up.
The prepare of students in your class aged \(40\) years is the empty prepare.
The set of prime numbers, which are irrational numbers, is the empty fix.
Each of the above sets is an empty set. Hence, the primal number for each of them is \(0\) (zero).
Singleton Set
A prepare containing one element is chosen a singleton (or unit) prepare.
Example: \(\left\{ { – 3} \right\}\)
\(\{ x:x \in W,~10 < one\} \)
\(\{ x:ten\) is capital letter of India \(\} \)
Each one of these is a singleton set. Hence, the cardinal number for each of them is \(i\) (one).
Equal Set up
Two sets are called equal sets if they have the same elements.
If sets \(A\) and \(B\) are equal, we write \(A=B\); and if sets \(A\) and \(B\) are not equal, nosotros write \(A≠B\).
Example: If \(A = \left\{ {~a,~b,~c} \right\}\) and \(B = \left\{ {~b,~a,~c} \correct\}\), then \(A=B\) considering the elements in a prepare tin can be repeated or rearranged. Hither, both sets \(A\) and \(B\) accept \(3\) elements each. Hence, their primal numbers are the same. \(north\left(A\correct) = north\left(B\right)\).
If \(A = \left\{ {~-3,~-2,~-1,~0,~1,~two,~3,~……} \right\}\) and \(B = \left\{ {x\mid x \in I,\, {x^2} < 10} \correct\}\), then \(A=B\) because if we write \(B\) in tabular form, we get the same elements. Here, both sets \(A\) and \(B\) have \(vii\) elements each. Hence, their cardinal numbers are the aforementioned. \(north\left(A\right) = north\left(B\right)\).
Equivalent Set
Two (finite) sets are called equivalent sets if they have the same number of elements. Thus, two finite sets, \(A\) and \(B\) are comparable sets, written as \(A↔B\) (read equally \('A\) is equivalent to \(B'\)), if \(northward\left(A\correct) = n\left(B\correct)\).
Example: If \(A = \left\{ {~a,~due east,~i,~o,~u } \right\}\) and \(B = \left\{ {~one,~ii,~3,~four,~5} \right\}\), then \(n(A)=v=due north(B)\).
Hence, the cardinal number of equivalent sets are equal.
Cardinal Number Formulas
- For two disjoint sets, \(n(A \cup B) = n(A) + n(B)\)
- For 2 overlapping sets, \(n(A \loving cup B) = n(A) + north(B) – n(A \cap B)\)
- For iii disjoint sets, \(n(A \cup B \cup C) = due north(A) + n(B) + north(C)\)
- For three overlapping sets, \(n(A \cup B \cup C) = n(A) + n(B) + n(C) – n(A \cap B) – n(B \cap C) – northward(C \cap A) + n(A \cap B \cap C)\)
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Solved Examples on Cardinal Numbers
Q.i. Write the given set in set-architect class: \(\left\{ {0,~1,~4,~9,~16,~25,~36} \correct\}\) .
Ans: The given numbers are a perfect square of the integers from \(-6\) to \(+six\).
\(\therefore \) Given set up \( = \{ x:x = n2,~\,due north \in I\) and \( – vi \leqslant n \leqslant + 6\} \).
Q.two. Write the given prepare in prepare-builder form: \(\left\{ {-10,~-5,~0,~five,~10,~fifteen,~twenty,~25} \correct\}\).
Ans: The given numbers are multiples of \(five\) lying betwixt \(-10\) and \(25\) (both inclusive).
\(\therefore \) Given set \( = \{ x|x = 5n,~\,n \in I\) and \( – 2 \leqslant n \leqslant 5\} \).
Q.3. Write the given set in roster form and find the cardinal number.
\(A\)={ vowels in the discussion TELEVISION}
Ans: Clearly, Television set has \(two Es, 2 Is\) and one \(O\), but while listing out elements in the roster form, we do not repeat elements.
In roster form, \(A = \left\{ {E,~\,I,\,~O} \right\}\) and \(n\left( A \correct) = 3\)
Q.4. Write the given gear up in roster form and find the cardinal number: \(B=\){ factors of \(24\)}
Ans: We know that,
The factors of \(24\) are \(1, two, 3, 4, 6, viii, 12, 24\)
In roster form, \(B = \left\{ {~1,~two,~three,~4,~6,~8,~12,~24} \right\}\). Since it has eight elements, \(n\left( B \right) = 8\).
Q.5. Write the given fix in roster form and notice the key number:
\(C = \{ x|x = 4n + 3,~\,0 < due north < seven,~\,n \in Due north\} \)
Ans: To get the elements of this fix, we shall have to substitute \(n=one, 2, three, four, 5\) and \(six\) in the given equation and list out all the values of \(x\) that we get.
For \(n=one, 10=4×one+3=7\);
For \(n=2, x=iv×2+3=11\), and then on.
Thus, we have \(C = \left\{ {~vii,~11,~fifteen,~xix,~23,~27} \right\}\) and \(northward\left( C\right) = 6\).
Summary
In this article, we discussed the definition of a fix, different means a prepare can be represented, the definition of central number and the cardinal number for different types of sets. Afterward reading this article, the students will have a fair understanding of the cardinal number related to the ready theory.
Ofttimes Asked Questions (FAQs) on Primal Numbers
Q.1. Is \(21\) a primal number?
Ans: Cardinal numbers are the numbers that are used for counting different things. These are also known as cardinals. These are whole numbers that start from \(0\) and goes on sequentially and are not fractions. So, yeah, \(21\) is the key number.
Q.2. What is the ordinal number of \(100\) ?
Ans: An ordinal number defines the position of an item or a number in a series, such equally 'first', '7th', 'eleventh' etc. Hence, the ordinal number of \(100\) is written every bit "one hundredth" or "the hundredth".
Q.iii. What is the Central number of a nil prepare?
Ans: A null set does not comprise any element in information technology. Hence, the primal number of a null set is \(0\) (zip).
Q .4. Is \(11\) a cardinal number?
Ans: Primal numbers are the numbers that are used for counting different things. These are also known as cardinals. These are whole numbers that beginning from 0 and goes on sequentially and are not fractions. So, yes, \(11\) is the cardinal number.
Q.v. What does Central number mean in math?
Ans: In mathematics, cardinal numbers are a generalization of the whole numbers used to measure the cardinality (size) of sets.
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