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.

In mathematics, a set is a collection of elements where an objective criterion determines whether a given object belongs to the group or not.

Example: You cannot define the set of tall people in a school without first clarifying what "tall" means. Does it refer to individuals over 170 cm? Or perhaps 180 cm? Establishing a clear, objective criterion is essential.

How do you represent a set?

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.

To define the set of even numbers between 1 and 10 (inclusive), we can use the characteristic property, which is a rule that determines which elements belong to the set and which do not.

For instance, we can express the even numbers from 1 to 10 as:

$$ P = \{x \in \mathbb{N} \mid x \text{ is even and } 1 \leq x \leq 10\} $$

Here, the symbol \( \mid \) (or \( : \), depending on preference) means "such that," and the logical condition states that \( x \) is both an even number and falls within the specified range.

Sets can also be visualized using Euler-Venn diagrams.

In these diagrams, each set is represented by a closed, non-overlapping curve that encloses points corresponding to the set's elements.

Euler-Venn diagrams are particularly useful for visually illustrating relationships between sets, such as intersections, unions, and differences.

That said, they become less practical when working with more complex sets or those with a large number of elements.

What’s the best method? The choice of the most suitable approach depends on the complexity of the set and the goal of the analysis. Each method has its strengths and limitations, making it more effective in certain situations. The roster method is simple and easy to use for small sets, but it becomes impractical for larger or infinite sets. The characteristic property offers a concise way to describe sets, especially infinite ones, but it requires a logical formula to define membership, which can sometimes be challenging. Moreover, it demands a solid understanding of mathematical language, making it less intuitive. Euler-Venn diagrams provide an intuitive visual representation of relationships between sets, but they work best for a small number of sets or straightforward relationships.

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. $$ vowels = \{ \ a \ , \ \color{red}a \ , \ e \ , \ i \ , \ o \ , \ u \ \} $$
• 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. $$ vowels = \{ \ u \ , \ o \ , \ i \ , \ e \ , \ a \ \} $$ $$ vowels = \{ \ i \ , \ u \ , \ e \ , \ a \ , \ o \ \} $$ $$ vowels = \{ \ e \ , \ a \ , \ u \ , \ o \ , \ i \ \} $$ 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?

There are two primary types of sets:

• Finite Sets
  A finite set contains a fixed number of elements. For example, the set of letters in the English alphabet is finite because it consists of exactly 26 letters. $$ A = \{ \ a \ , \ b \ , \ c \ , \ d \ , \ e \ , \ f \ , \ g \ , \ h \ , \ i \ , \ j \ , \ k \ , \ l \ , \ m \ , \ n \ , \ o \ , \\ \ \ \ \ \ \ \ \ \ \ \  \ p \ , \ q \ , \ r \ , \ s \ , \ t \ , \ u \ , \ v \ , \ w \ , \ x \ ,  \ y \ , \ z \ \} $$
• Infinite Sets
  An infinite set contains an unlimited number of elements. For instance, the set of positive integers $ \mathbb{Z} $ is infinite.

If a set \( A \) has a finite number of elements, the total number is referred to as the set's cardinality, represented by the symbol \( |A| \).

For example, take the set \( V = \{ a, e, i, o, u \} \), representing the vowels of the alphabet. This set has five elements, so its cardinality is \( |V| = 5 \). On the other hand, the set of letters in the English alphabet has a cardinality of \( |A| = 26 \), as it contains twenty-six elements.

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.