9. Sequences, Cartesian product of sets

Sequences

Sequence

A sequence is a list of things, taken in a certain order. $$ \begin{array}{c} (1, 2, 3, -3, 8) \\ (\text{Among, Us, Impostor}) \end{array} $$

Unlike sets, they are instead denoted using brackets instead of braces.

Important features of sequences: ?

• Order of list matters. $(1,2,3)$ and $(3,2,1)$ are different sequences.
• Repeated occurrences matter. $(H,E,L,O)$ and $(H,E,L,L,O)$ are different sequences.

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Tuple

A tuple is a finite [[Sequence|sequence]].

Two tuples are equal if they have the same length and their corresponding elements are the same:

$$\boxed{(x_1,x_2,...,x_k)=(y_1,y_2,...,y_n)}\text{ means that }k=n\text{ and that }{x}_1=y_1\text{ and so on...}$$Link to original

k-Tuple

A $k\text{-tuple}$ is a sequence with $k$ elements, the number $k$ is called the length of the tuple.

Ordered pair

An ordered pair is another name for a $2\text{-tuple}$, $$ \boxed{(a, b) = (c, d)} \text{ means that } a = c \text{ and } b = d $$

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Cartesian product of sets

Cartesian product of sets

The Cartesian product of sets $A$ and $B$ is the set $$ A \times B = \{ (x, y) | x \in A \text{ and } y \in B \} $$

$A\times B$ consists of ordered pairs $(x,y)$ where $x\in A$ and $y\in B$.

For example, let $A=\{1,2\}$ and $B=\{a,b,c\}$. Hence, $A\times B=\{(1,a),(2,a),(1,b),(2,b),(1,c),(2,c)\}$.

Generic Format

The Cartesian product of sets $A_1,A_2,...,A_k$ is the set $$ A_1 \times A_2 \times ... \times A_k = \{ (x_1, x_2, ..., x_k) | x_i \in A \text{ for } i = 1, 2, ..., k \} $$

$A_1\times A_2\times ...\times A_k$ consists of those $k\text{-tuples }(x_1,x_2,...,x_k)$ where $x_1\in A_1,x_2\in A_2,..$

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