Tuples

• a set contining no elements is the empty set.

• a set containing 1 element is a singleton set, or unit set

• a set containing 2 elements is a pair, unordered pair, 2-tuple

• a set containing $k$ elements is a k-tuple

A set $X$ is called an ordered pair if $X = \{\{x\},\{x,y\}\}$ for some $x,y$. This is commonly abbreviated by writing an ordered pair as $\langle x,y \rangle$ or $\{(x,y)\}$.

Two ordered pairs are equal if their respective components are the same: let $(a, b)$ and $(c, d)$ be ordered pairs; then $(a, b) = (c, d)$ iff $a = c$ and $b = d$.

An ordered pair $\langle x, x \rangle$ is $\{\{x\},\{x, x\}\}$ by definition, which first collapses into $\{\{x\},\{x\}\}$ and then into $\{\{x\}\}$. Therefore $\langle x, x \rangle = \{\{x\}\}$.

https://en.wikipedia.org/wiki/Tuple

A set containing two elements may be referred to as an unordered pair.

An ordered pair is a pair of objects, $(a,b)$, where the object $a$ is called the first entry, the object $b$ is the second entry of the pair. Unlike unordered pairs (i.e. sets with two elements), an ordered pair is affected by the order of its two elements, so $(a,b)\neq (b,a)$.

Two ordered pairs are equal if their respective components are the same: let $(a, b)$ and $(c, d)$ be ordered pairs; then $(a, b) = (c, d)$ iff $a = c$ and $b = d$.

The Cartesian product of two sets $A$ and $B$ is another set, denoted as $A\times{B}$ and defined as $A\times{B} = \{(a,b) : a\in A, b\in B\}$

Thus $A\times{B}$ is a set of all ordered pairs of elements from $A$ and $B$.

If $A$ and $B$ are finite sets, then $|A\times B| = |A|*|B|$

The Cartesian (cross) product A × B of two sets is defined as A × B = {(a, b) : a ∈ A, b ∈ B}

So, the × operation pairs the elements of A with the elements of B in such a way that the elements of A appear as first components, and the elements of B appear as second components.

It is also possible to define Cartesian products for more than two factors, in which case we would not have a pair i.e. 2-tuple, but 3-tuple. Even so, all the n-tuples can be represented by an ordered pair (recursively), e.g. an 4-tuple, $(a,b,c,d)$, can be represented as $(a,(b,(c,d)))$.