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 elements is a k-tuple
A set is called an ordered pair if for some . This is commonly abbreviated by writing an ordered pair as or .
Two ordered pairs are equal if their respective components are the same: let and be ordered pairs; then iff and .
An ordered pair is by definition, which first collapses into and then into . Therefore .
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A set containing two elements may be referred to as an unordered pair.
An ordered pair is a pair of objects, , where the object is called the first entry, the object 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 .
Two ordered pairs are equal if their respective components are the same: let and be ordered pairs; then iff and .
The Cartesian product of two sets and is another set, denoted as and defined as
Thus is a set of all ordered pairs of elements from and .
If and are finite sets, then
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, , can be represented as .
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