Set theory is a subfield of mathematics in which we study the characteristics of various sets and the sets themselves. A collection of different things is what we mean when we talk about a set. The elements of a specific set are the things that make up that set. The study of such sets, as well as the relation that relates the sets to the components they include, is known as set theory. Sir Georg Ferdinand Ludwig Philipp Cantor, a German mathematician, is credited with being the first person to develop the concept of set theory. The development of set theory followed a path very distinct from that of the majority of other subfields of mathematics over its entire history. It was devised so that people may have conversations about groups of things that together stand for a certain category. The study of sets has shown to be an exceptionally useful tool in defining a variety of the most difficult and crucial mathematical structures.
A finite set is a countable set having a finite number of items. An infinite set, on the other hand, is made up of an infinite number of components and can be counted or uncounted. Yes, countable and uncountable do not imply finite and infinite sets. There is a distinction to be made. Sets like N (natural numbers) and Z (integers), for example, are countable even though they are infinite since they may be listed. In other words, each of these sets has a one-to-one correspondence (bijection) with the set of natural numbers N, and hence they are countable. The set of all real numbers R, on the other hand, is uncountable because we can’t list its elements and hence there can’t be a count.
In General, if there is a bijection from A to B, two sets A and B are said to have the same cardinality.
Cardinality Of Finite Set
To comprehend the cardinality of finite sets, you must first comprehend finite sets. If a set has exactly n unique items and n is a non-negative integer, then is said to be finite. Numerable sets are sometimes known as finite sets. Then cardinality of sets is defined as n.
The symbols denoting the cardinality of sets are |A|, card (A), and n. (A). The most common representations are |A| and n. (A). It is said to be finite if a set is finite.
- It is an empty set, or,
- if there is one to one correspondence between the rudiments in the set
Symbol Related to Cardinality of Set
Symbol | Symbol Name | Meaning |
a∈B | element of | set class |
(a,b) | ordered pair | collection of 2 elements |
x∉A | not element of | no set class |
|B|, #B | cardinality | the number of elements of set B |
Properties
Now that we’ve learned what the cardinality of a set means, let’s look at some of its key qualities to help us better comprehend the notion.
- If A and B are two disjoint sets, also n(A U B) = n(A) + n (B).
- If there are 2 sets given A and B, also n (A U B) = n(A) + n (B) – n (A ∩ B).
This principle is known as “inclusion-exclusion principle”.
- The relation of sets having the same cardinality is an equivalence relation.
- A set A is countable if it is either finite or there is a bijection from A to N.
- N, Z, and Q are countable.
- R is uncountable.
- For Any given subset of a countable set is countable.
- If A and B are countable then their cartesian product A X B is also countable.
Conclusion
The number of individual components that make up a finite set is used as a measurement for determining the set’s size, which is referred to as its cardinality. It is important to keep in mind that constructing a correspondence of 1-1 between the items of a set and the numbers in the sequence 1,2,…,n is equivalent to counting the number of elements in the set.