3.1. Cardinality of Finite Sets

So let us begin with the cardinality of finite sets first. So if I ask you what is the cardinality of this set X which consists of the elements Ram, Sham, Gita and Sita. You will say its cardinality is 4 because it has 4 elements.

Cardinality refers to the number of elements in a set. For example, in the set X containing Ram, Sham, Gita, and Sita, we can observe that there are exactly 4 unique elements. Thus, we say the cardinality of set X is 4. Cardinality helps us quantify how many items are present in any given set, which is fundamental in mathematics and set theory.

Think of a fruit basket that contains exactly four apples. To understand the cardinality of this basket, you can count the apples one by one: apple 1, apple 2, apple 3, and apple 4. In this case, just like the set X, the basket contains 4 specific items, thus the cardinality is 4.

Another way to put it is as follows: We can say that the cardinality of the set X is 4 because there is a bijection between the set X and the set consisting of the elements 1, 2, 3, 4.

A bijection is a one-to-one correspondence between two sets. If we can pair every element of set X distinctly with the numbers 1, 2, 3, and 4, we establish that both sets have the same cardinality. For example, if Ram maps to 1, Sham to 2, Gita to 3, and Sita to 4, we have formed a perfect bijection, thereby confirming that both sets have 4 elements.

Imagine assigning a unique seat number to each student (Ram, Sham, Gita, and Sita) in a classroom. Just like how every student gets a unique seat number (1 to 4), a bijection pairs each element of set X with a unique number. This highlights the relationship between counting the number of items (students) and establishing their position (seat numbers).

Now due to the same reason if I consider another set Y consisting of the elements Delhi, Kolkata, Mumbai, and Chennai, its cardinality is also 4.

The cardinality of set Y, which includes four cities - Delhi, Kolkata, Mumbai, and Chennai - is also 4. Similar to set X, if we can create a bijection between set Y and the set of numbers 1 to 4 (where each city is matched to a unique number), it reinforces the concept of cardinality and helps us recognize that different sets can have the same number of elements.

Consider a group of four friends visiting four different attractions. Each attraction can be numbered, just like the cities in set Y. By pairing each friend with a unique attraction number, we can easily say that the 'group of attractions' has a cardinality of 4, just matching the cardinality we found with the group of cities.

Now how do we compare the cardinality of finite sets; say if I am given the set X and the set Y. It is easy to see that the set X its cardinality is less than cardinality of set Y.

When comparing two sets, if the number of elements in set X is less than that in set Y, we denote this relationship as |X| < |Y|. This relationship is crucial in understanding the sizes of different sets. We can also visualize this through injective functions, where every element in the smaller set can be mapped distinctly to an element in the larger set.

Imagine two bags of candies. If Bag X contains 3 candies and Bag Y contains 5 candies, it is clear that Bag X (|X| = 3) has a lesser quantity compared to Bag Y (|Y| = 5). This principle allows us to easily determine that the cardinality of Bag X is less than that of Bag Y.

So there can be many injective functions one possible objective mapping I can define as Ram getting mapped to Delhi and Shyam getting mapped to Kolkata.

An injective function is one where each element from set X can be matched with a unique element in set Y, without any overlaps. For instance, if we map students (Ram, Sham) to city names (Delhi, Kolkata), each student is associated with exactly one city, illustrating the concept of injective functions in comparing cardinalities.

Imagine a situation where every superhero can only have one unique sidekick. If Spider-Man is assigned to a specific city, and no one else can be assigned to that same city, this demonstrates an injective mapping. Just as in our mapping, each superhero (element of set X) has a unique city (element of set Y) tailored for them.

So based on this example I can give the following definition I can say that the cardinality of the set A is less than or the same as the cardinality of set B.

In set theory, we denote the cardinality relationship using the notation |A| ≤ |B|, which means that the number of elements in set A is less than or equal to that in set B. This notation simplifies the comparison of different sets in terms of their sizes and helps in understanding the ordering of finite sets.

If we compare two boxes of chocolates, say Box A with 4 chocolates and Box B with 6 chocolates. We can express this relationship with our notation: |A| ≤ |B| (4 ≤ 6). This comparison highlights the relationship in a straightforward way, making it easier to interpret.

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