Sequences or tuples

A sequence is an ordered collection of objects where the order of the elements matters.

Typically, sequences are represented with elements enclosed in parentheses. For example, the sequence of numbers 8, 12, and 7 is written as \( (8, 12, 7) \).

Sequences have two fundamental characteristics:

• Order of elements: The order in a sequence is crucial.

  For example, \( (1, 2, 3) \) is different from \( (3, 2, 1) \). Think of a password: reversing the characters renders it unusable.

• Repetition of elements: In a sequence, elements can repeat, and the repetition is significant.

  For example, \( (1, 1, 2, 3) \) is different from \( (1, 2, 3) \). Imagine writing the same word twice in a sentence: it changes the meaning.

As we will see, sets and sequences are often represented similarly but are not the same.

Types of sequences

Sequences can be finite or infinite:

• Finite sequences
  Have a limited number of elements. Here is an example of a finite sequence: \( (a, b, c) \).
• Infinite sequences
  Have an unlimited number of elements, often represented with ellipses. Think of natural numbers: they never end. For example, the infinite sequence of odd numbers is written as \( (1, 3, 5, \ldots) \).

Tuples or n-tuples

A finite sequence with \( n \) elements is called a tuple or n-tuple (ennupla).

Depending on the number of elements, a tuple is referred to as:

• 1-tuple (or singleton): A tuple with one element, e.g., \( (x) \).
• 2-tuple (or ordered pair): A tuple with two elements, e.g., \( (x, y) \).
• 3-tuple (or triple): A tuple with three elements, e.g., \( (x, y, z) \).
• k-tuple: A tuple with \( k \) elements.

For example, a 4-tuple contains exactly four elements, like \( (a, b, c, d) \).

The Cartesian Product and Sequences

The Cartesian product is a concept that creates ordered pairs or tuples by combining elements from two or more sets.

While it is not a sequence itself, the Cartesian product results in ordered collections of elements.

For instance, the Cartesian product of sets \( A \) and \( B \), denoted as \( A \times B \), consists of all ordered pairs \((a, b)\) where \( a \in A \) and \( b \in B \).

These ordered pairs can be regarded as sequences of length two.

Another practical example of ordered pairs is the coordinates of points (x,y) on the Cartesian plane.

As you can easily see, the point at coordinates (1,2) does not occupy the same position as the point at coordinates (2,1).

Differences between sets and sequences

The main difference between sets and sequences is the importance of the order and repetition of elements:

In a set, order and repetition do not matter.

For example, the set of numbers \( \{a, b, c\} \) remains unchanged regardless of the order \( \{b, a, c\} \) or \( \{c, a, b\} \). When collecting fruit in a basket, the sequence doesn't matter, but the content does.

In a sequence, however, order and repetition are fundamental.

For example, the sequences \( \{a, b, c\} \) and \( \{b, a, c\} \) are different. It's like following instructions to assemble furniture; they must be followed in a precise order. To bake a cake, you need to follow a sequence, not just a random set of ingredients!

In conclusion, while sequences and sets are often represented similarly with parentheses, their meanings are entirely different.