Cardinal Numbers: Definition, Examples, and Sample Questions

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Cardinal numbers are those that tell us how many of anything there are, such as five, seven, eight, 10, and so on. Simply said, cardinal numbers respond to the question "How many?" Cardinal numbers can't help but be employed as counting numbers. Ordinary numbers of cardinals are the terms used to describe them. A lot of cardinal numbers start at 1 and go all the way up to infinity. To respond to the question "what number of?" we employ cardinal numbers. For instance, how many understudies are attending the school barbecue? Any number, such as 20, 23, 30, and so on, might be the reaction. As a result, this huge number belongs to the category of cardinal numbers. Let’s discuss cardinal numbers in detail along with some important questions related to it.

Table of Content

What are Cardinal Numbers?
Examples of Cardinal Numbers
Difference between Cardinal Numbers and Ordinal Numbers
Cardinality
Things to Remember
Sample Questions

Key Terms: Nominal, Cardinality, finite and infinite, Decimals

What are Cardinal Numbers?

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Cardinal numbers are an extension of natural numbers since they include all counting numbers beginning with one. Cardinals are sometimes known as cardinal numbers. The word cardinal numbers or cardinal was invented to express the cardinality (size) of sets, which is the size of a set. The number of items in a finite set is called cardinality, and it is used to characterize the size of the sets. Consider the following two sets: Set A = 2, 4, 6, 8 and Set B = 1, 2, 3, 5, 7. Set A has a cardinality of four because there are four elements in it, but set B has a cardinality of five since there are five items in it.

Examples of Cardinal Numbers

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The cardinality of a group is the number of items that may be found inside that group.

In the kitchen, there are five cups.
Four trucks are sharing a lane.
Hritik has three parrots and two dogs as pets.

The cardinal numbers in the preceding three cases are 5, 4, 3, and 2. So, in essence, it represents the quantity of anything, regardless of its arrangement. It specifies the size of a set but does not take into consideration its order.

The natural numbers that determine cardinality make up the set of finite numbers. The set of infinite cardinals, on the other hand, describes the size of infinite sets. There are no fractions or decimals for the cardinals; they merely count numbers.

Read More: Set Theory

Difference between Cardinal Numbers and Ordinal Numbers

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Cardinal Numbers:

Cardinal is a utility function that identifies an individual's pleasure with a commodity and may be backed up by a numeric number.
The quantitative method is described by the term "cardinal utility."
Cardinal utility measurement is less practical because it is objective in nature.
Cardinal utility is represented by numbers that finish in one, two, three, and so on.
Based on utilities, utility measurement is carried out.
Mina, for example, receives 70 utils of satisfaction from coffee, but she receives just 30 utils from tea.

Ordinal Numbers:

Ordinal utility refers to the fact that user goods can be ordered or positioned in order of choice but not quantitatively appraised.
The qualitative approach is described by the term "ordinal usefulness."
Ordinal is a subjective utility measurement, making it more practical and reasonable.
The ordinal utility - represented by integers ending in 1-st', 2-nd', and 3-rd' - is critical.
The ranking system is used to assess performance.
Mina says that she prefers coffee to tea for its fulfillment.

Cardinality

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The cardinality of a set (group) indicates how many items or concepts are contained within it.

Example: What is the cardinality of the flowers in the vase? In this vase, there are five flowers. As a result, flowers have a cardinality of 5.The first cardinal number is 1. Fractions and decimals indicate a portion of a whole or a group (less than one). Fractions and decimals are thus not cardinal numbers

Things to Remember

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Natural numbers are sometimes known as cardinal numbers. A set of entire numbers is made up of Natural Numbers (Cardinal Numbers) and 0 (zero).
One, two, three, and so on are all examples of cardinal numbers. The number of items is indicated by the cardinal number, whereas the number of items is shown by the ordinal number.
Decimals are mathematical numbers that are divided into two parts: a whole number component and a fractional part separated by a decimal point. 10.2 is a decimal number, for example. For decimals, use the cardinal number: 2.7 two-point seven, 5.36 five-point-three-six.
The cardinal number in the context of a set is the total number of items in it. In other terms, the cardinal number of a set is the number of different items contained in the set. The cardinal number of a set A is denoted by the letter n. (A). Because there are 5 elements in the set W = 1, 3, 5, 7, 9, the cardinal number is n(W)=5.
A null set is one that has no elements. As a result, a null set's cardinal number is 0. (zero).
Cardinal numbers can be used to symbolise a collection of ordinal numbers. Counting is always done using cardinal numbers, which are expressed as 'how many'.
Cardinal numbers are not fractions or decimals. Zero (0) isn't a cardinal number because it has no meaning.
The cardinality of a set is the number of objects or items in the set.

Read More: Centroid of a Triangle

Sample Questions

Ques: Is the number 21 a cardinal number? (1 mark)

Ans: Cardinal numbers are the numbers that are used to count various objects. Cardinals are another name for them. These are full numbers, not fractions, that start at 0 and move up consecutively. So, indeed, the cardinal number is 21.

Ques: In a class of 50 pupils, 10 kids did not choose math, 15 students did not choose science, and two students did not choose either. How many kids in the class choose math and science as their subjects? (2 marks)
(a) 24
(b) 25
(c) 26
(d) 27

Ans: (d) 27

Explanation: Total students = 50

Students who did not opt for math = 10

Students who did not opt for science = 15

Students who did not opt for either maths or science = 2

A total of 40 students in math and 13 did not opt for science but did for math = 40 – 13 = 27

So, students of the class opted for both math and science are 27

Ques:In a group of 80 people, 37 like cold drinks and 52 like hot drinks and each person likes at least one of the two drinks. Find How many people like both coffee and tea? (3 marks)

Ans: Let A = Set of people who like cold drinks.

B = Set of people who like hot drinks.

Given

(A ∪ B) = 80 n(A) = 37 n(B) = 22 then;

n(A ∩ B) = n(A) + n(B) – n(A ∪ B)

= 37+52-80

= 89 – 80

= 9

Therefore, 9 people like both tea and coffee.

Ques:N and M elements are found in two finite sets. The total number of elements in the first set's power set is 48 greater than the total number of elements in the second test's power set. Then the value of M and N are (4 marks)
(a) 7, 6
(b) 6, 4
(c) 7, 4
(d) 6, 3

Ans:(b) 6, 4

Explanation: Let A and B be two sets having m and n numbers of elements respectively

Number of subsets of A = 2m

Number of subsets of B = 2n

Now, according to question

2m – 2n = 48

⇒ 2n(2m – n – 1) = 24(22 – 1)

So, n = 4

and m – n = 2

⇒ m – 4 = 2

⇒ m = 2 + 4

⇒ m = 6

Ques: What exactly are the distinctions between cardinal, ordinal, and nominal numbers? (5 marks)

Ans: The decimal number system defines cardinal numbers, ordinal numbers, and nominal numbers. Cardinal numbers are used to count, as the name implies. Ordinal numbers are used to group numbers together, whereas nominal numbers are used to identify people. A person's passport number is an example of a nominal number. Let's examine the differences between the three sorts of numbers.

Cardinal Numbers: Cardinal numbers are the counting numbers. It assists us in determining the number of components present.

Because 0 is not a counting number, the least cardinal number is 1

1,2,3,4, and so on are examples of cardinal numbers.

Ordinal Numbers: Ordinal numbers are a type of natural number that is used to indicate how different elements are arranged.

It essentially defines an element's location in relation to other elements.

1st-first, 2nd-second, 12th-twelfth, and so on are examples of ordinal numerals.

Nominal Numbers: Nominal numbers are numbers that are used to identify anything.

These are used to identify individual items.

Passport numbers, cell phone numbers, ZIP code numbers, and other nominal numbers are examples.

Ques: Every student at a school plays either hockey or football, or both. There are 400 people who play football, 150 people who play hockey, and 130 people who play both games. Calculate (3 marks)
(i) The number of pupils who play Football only,
(ii) The number of pupils who play Hockey only,
(iii) The total number of pupils in the school.

Ans:H = Hockey and F = Football

n (H ) = 150 n (F)= 400

n ( H ∩ F) = 130

(i) The number of pupils who only play Football = n (F – H )

n (F – H ) = n(F) – n( F ∩ H )

= 400 – 130

= 270

(ii) The number of pupils who only play Hockey = n (H – F )

n (H– F ) = n(H) – n( F ∩ H )

= 150 – 130

= 20

(iii) The total number of pupils in school

= n(H) + n(F) – n (F ∩ H)

= 150 + 400 – 130

= 420

Ques:What is 100's ordinal number? (2 marks)

Ans: An ordinal number, such as 'first, “seventh,' 'eleventh,' etc., denotes the position of an item or a number in a sequence. As a result, "one hundredth" or "the hundredth" is written for the ordinal number 100.

Ques:What is the difference between natural and cardinal numbers? (3 marks)

Ans: Cardinal numbers and natural numbers are both counting numbers that begin with 1. The magnitude difference between cardinal and natural numbers is the sole distinction. Natural numbers are finite numbers that make up a subset of cardinal numbers, whereas cardinal numbers are infinite. The number 0 is included in natural numbers; however, it is not the same as the cardinal numbers. The number 0 is not utilised in the cardinal number.

Ques: { (A, B) : A² +B² = 1} on the sets has the following relation (2 marks)
(a) reflexive
(b) symmetric
(c) none
(d) reflexive and transitive

Ans: (b) symmetric

Explanation: Given {(a, b) : a² + b² = 1} on the set S.

Now a² +b² = b² + a² = 1

So, the given relation is symmetric

Ques:Let A and B be two finite sets such that n(A) = 30, n(B) = 18 and n(A ∪ B) = 26, find n(A ∩ B)? (2 marks)

Ans: Formula for n(A ∪ B) = n(A) + n(B) – n(A ∩ B).

Rearranging it we get the n(A ∩ B) = n(A) + n(B) – n(A ∪ B)

=30+18 – 26

= 22

Therefore, n(A ∩ B) = 22.

Mathematics Related Links:
Sets	Quadratic equations	Statistics
Limits	Probability	Lines
Binomial Theorem	Combination	Linear inequalities
Complex Numbers	Trigonometric	Relations

Cardinal Numbers: Definition, Examples, and Sample Questions

What are Cardinal Numbers?

Examples of Cardinal Numbers

Difference between Cardinal Numbers and Ordinal Numbers

Cardinal Numbers:

Ordinal Numbers:

Cardinality

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

2.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

3.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

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Cardinal Numbers: Definition, Examples, and Sample Questions

What are Cardinal Numbers?

Examples of Cardinal Numbers

Difference between Cardinal Numbers and Ordinal Numbers

Cardinal Numbers:

Ordinal Numbers:

Cardinality

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

2.Find:Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

3.If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

4. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

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1.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

2.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

3.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).