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Different items are counted using cardinal numbers. Cardinals are another name for them. These are natural numbers that begin with 1 and continue in order, and are not fractions. The term “cardinality” refers to the number of things that exist in a group. For example, if we want to count the quantity of apples in the basket, we must use numbers like 1,2,3,4,5,… and so on. Numbers aid in the counting of objects, such as the number of persons present in a location or a group.
Definition of cardinal numbers: The symbol n is used to signify the number of items in a set A, n(A). It’s referred to as a set A cardinal number.
It’s possible that a set has
1. There are no elements
2. At least one element with
a. There are a finite number of elements.
b. There is no limit to the number of elements.
Cardinal Numbers for Different Sets
The cardinal number in the context of a set is the total number of elements in it. In other terms, the cardinal number of a set is the number of different items present 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.
Different types of sets are defined as follows, based on the number of elements in the set:
Finite set: A finite set is one that contains a finite (counted) number of different elements. To put it another way, a set is considered a finite set when the counting of its various parts reaches a conclusion.
Example, A= { a, b, c, d, e} is a five-element finite set. Set A’s cardinal number, indicated by n(A), is 5.
Infinite set: An infinite set is one that has an infinite (uncountable) number of different elements. To put it another way, a set is termed when the counting of its various elements does not finish.
N= 1, 2, 3, 4, …..is a collection of natural numbers.
W={ 0, 1, 2, 3, …..} is a set of whole numbers.
{ -3, -2, -1, 0, 1, 2, 3, ……} is a set of integers.
Each of the sets shown above is an infinite set. As a result, the cardinal number for each of them is infinite.
Empty set: A blank, void, or null set is one that has no elements. It is represented by the symbol ϕ or { }.
The empty set is made up of cats with two tails.
The empty set is the group of pupils in your class that are 40 years old.
The empty set is made up of prime numbers, which are irrational numbers.
Each of the sets listed above is empty. As a result, each of them has a cardinal number of 0. (zero).
Singleton set: A singleton (or unit) set is a set with only one element.–3}
{x:xW, x<1}
Each of these is a one-of-a-kind set. As a result, each of them has a cardinal number of one (one).
Equals Set: When two sets have the same elements, they are referred to as equal sets.
We write A=B if sets A and B are equal, and we write AB if they are not equal.A≠B.
Because the members in a set can be repeated or rearranged, A={ a, b, c} and B={ b, a, c}, Both sets A and B contain three elements. As a result, their cardinal numbers are identical. n(A)=n(B).
Equivalents set: If two (finite) sets have the same number of items, they are called equivalent sets.
If, A={ a, e, i, o, u} and B={ 1, 2, 3, 4, 5}, n(A)=5=n(B).
Cardinal and Ordinal Numbers: What’s the Difference?
Cardinal numbers are all natural numbers that are not divisible by two. Counting is done with cardinal numbers. An ordinal number, on the other hand, is a number that indicates the position or location of an object. 1st, 2nd, 3rd, 4th, 5th, and so on. For ranking, ordinal numbers are utilized.
Cardinal numbers formula
For 2 disjoint sets, n(AB)=n(A)+n(B)
For 2 overlapping sets, n(AB)=n(A)+n(B)–n(AB)
For 3 disjoint sets, n(ABC)=n(A)+n(B)+n(C)
For 3 overlapping sets, n(ABC)=n(A)+n(B)+n(C)–n(AB)–n(BC)–n(CA)+n(ABC)
Conclusion
The definition of a set, different ways a set can be represented, the definition of a cardinal number, and the cardinal number for different types of sets They’re also known as Cardinals. These are not fractions, but rather natural numbers that start with 1 and go up in order. The phrase “cardinals” refers to the number of items in a collection that were all explored in this article.
