1tuple

A 1-tuple is an ordered tuple with exactly one element. In mathematics, a k-tuple is typically defined as a function from the index set {1, 2, ..., k} to a target set X; thus a 1-tuple over X is a function from {1} to X. The value of that function is the sole coordinate of the tuple. Consequently, the collection of all 1-tuples over X is naturally in bijection with X itself.

Because a 1-tuple contains only one element, many authors identify a 1-tuple with its single component. Formally, however, a 1-tuple preserves the notion of a tuple as an ordered collection, even when that collection has length one. This helps unify definitions of n-tuples for n ≥ 1 and supports basic constructions like Cartesian products and multi-variable mappings.

In practice, a 1-tuple (a) is the single element a drawn from X. For example, if X = {0, 1}, the 1-tuples are (0) and (1). From a set-theoretic perspective, describing (a) as a function f with f(1) = a highlights its nature as a single-coordinate object rather than just an element.

Programming languages sometimes treat 1-tuples differently. In Python, for instance, (a) is not a 1-tuple but a parenthesized expression equal to a, while (a,) denotes a true 1-tuple. This distinction is a common source of confusion for beginners.

The concept of a 1-tuple underpins the general theory of tuples and is a foundational case in discussions of n-tuples, Cartesian products, and coordinate representations in mathematics and related fields.