Sets
Imagine a set as a collection, much like a basket filled with distinct objects, known as elements or members, that all share a common property.
These components within the basket or "set" are aptly referred to as its elements.
Before you can define a set, you first need to lay down what we call a membership criterion.
What exactly is this? It's a rule, plain and simple, one that helps to unambiguously determine whether an object belongs to the set in question or not.
One way to formally describe a set is by using roster notation, where the elements of the set are listed inside braces. This notation is particularly useful and straightforward when dealing with finite sets.
For instance, consider the set of vowels. It includes the letters a, e, i, o, u:
$$ vowels = \{ \ a \ , \ e \ , \ i \ , \ o \ , \ u \ \} $$
In this case, the letters are the elements of the set "vowels".
This is known as a finite set as it contains a finite number of elements.
Another example is the set of even numbers between 1 and 10.
$$ P = \{ \ 2 , 4, 6, 8, 10 \ \} $$
Here, the even numbers are the elements of the set P.
A noteworthy aspect of the membership criterion is its objectivity. It must allow anyone to clearly determine whether an object belongs to the set or not. For example, "funny films" can't form a set since humor is subjective - what you find amusing might not be the same for others.
Let's delve into the characteristics of a set
Two main attributes define a set:
- Uniqueness of elements
Within a set, each element is distinct and cannot be duplicated. An element, once it belongs to a set, can't appear multiple times in the same set. For instance, you can't include the letter 'a' twice in the set of vowels. $$ vocali = \{ \ a \ , \ \color{red}a \ , \ e \ , \ i \ , \ o \ , \ u \ \} $$ - Irrelevance of order
Sets do not have an inherent order to the elements. The sequence of elements doesn't affect the set. For instance, you could write the set of vowels in numerous ways and it would remain the same set since the elements within are identical. $$ vocali = \{ \ u \ , \ o \ , \ i \ , \ e \ , \ a \ \} $$ $$ vocali = \{ \ i \ , \ u \ , \ e \ , \ a \ , \ o \ \} $$ $$ vocali = \{ \ e \ , \ a \ , \ u \ , \ o \ , \ i \ \} $$ Anche se la disposizione degli elementi cambia, è sempre lo stesso insieme perché gli elementi sono gli stessi.
What's the symbol for a set?
In mathematical notation, sets are generally denoted by uppercase alphabet letters (A, B, C, ...), while the elements are represented by lowercase letters (a, b, c, ...).
For instance, let's represent the set of vowels as A and the set of consonants as B $$ A = \{ \ u \ , \ o \ , \ i \ , \ e \ , \ a \ \} $$ $$ B = \{ \ b \ , \ c \ , \ d \ , \ f \ , \ g \ , \ h \ , \ j \ , \ k \ , \ l \ , \ m \ , \ n \ , \ p \ , \ q \ , \ r \ , \ s \ , \ t \ , \ v \ , \ w \ , \ x \ , \ z \ \} $$
The membership of an element in a set is represented by the symbol ∈
To illustrate, let's say the element 'a' belongs to set A $$ a \in A $$
If an element doesn't belong to a set, the non-membership symbol ∉ is used.
For example, the element 'b' doesn't belong to set A $$ b \notin A $$
What are the types of sets?
Typically, there are two primary categories of sets:
- Finite sets
A finite set has a specific number of elements. For example, the set of alphabet letters is a finite set. - Infinite sets
An infinite set has an infinite number of elements. For example, the set of positive integers is an infinite set.
A Practical Example
Set A is the set of European cities and B is the set of Italian cities.
$$ A = \{ \ Paris \ , \ Rome \ , \ Berlin \ , \ London \ , \ ... \} $$
$$ B = \{ \ Rome \ , \ Milan \ , \ Florence \ , \ Naples \ , \ ... \} $$
Let 'a' represent the city of Rome.
$$ a \ = \ Rome $$
The element 'a' (Rome) belongs to both set A (European cities) and set B (Italian cities)
$$ a \in A $$
$$ a \in B $$
Now let 'b' represent the city of Paris.
$$ b \ = \ Paris $$
The element 'b' (Paris) belongs to set A (European cities) but not to set B (Italian cities).
$$ b \in A $$
$$ b \notin B $$
Finally, let 'c' represent the city of New York.
$$ c \ = \ New \ York $$
The element 'c' does not belong to either set A (European cities) or set B (Italian cities)
$$ c \notin A $$
$$ c \notin B $$
By now, you should have a clear understanding of what a set is.
