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n-Tuple — Definition, Formula & Examples

An n-tuple is an ordered list of exactly n elements, written in parentheses and separated by commas. Unlike a set, the order of elements matters and repetitions are allowed.

An n-tuple is a finite ordered sequence (a1,a2,…,an)(a_1, a_2, \ldots, a_n) of nn elements, where each aia_i belongs to some specified set. Two n-tuples (a1,…,an)(a_1, \ldots, a_n) and (b1,…,bn)(b_1, \ldots, b_n) are equal if and only if ai=bia_i = b_i for every i∈{1,2,…,n}i \in \{1, 2, \ldots, n\}.

Key Formula

(a1,a2,…,an)(a_1, a_2, \ldots, a_n)
Where:
  • nn = The number of elements (length of the tuple)
  • aia_i = The element in the i-th position, where i ranges from 1 to n

How It Works

You build an n-tuple by choosing a value for each of its nn positions. A 2-tuple is called an ordered pair, a 3-tuple is an ordered triple, and so on. The key property is that position matters: (3,5)(3, 5) and (5,3)(5, 3) are different 2-tuples, even though they contain the same elements. Repeated entries are also meaningful — (2,2,7)(2, 2, 7) is a valid 3-tuple distinct from (2,7,7)(2, 7, 7). In linear algebra, vectors in Rn\mathbb{R}^n are represented as n-tuples of real numbers. In databases and programming, each row of a table is often modeled as an n-tuple.

Worked Example

Problem: Determine whether the 4-tuples (1, 0, 3, 5) and (1, 3, 0, 5) are equal.
Step 1: Compare the elements position by position.
a1=1=b1,a2=0,  b2=3a_1 = 1 = b_1,\quad a_2 = 0,\; b_2 = 3
Step 2: Since the second components differ (0≠30 \neq 3), the two 4-tuples are not equal. Order matters in a tuple, so swapping the 0 and the 3 creates a different object.
(1,0,3,5)≠(1,3,0,5)(1, 0, 3, 5) \neq (1, 3, 0, 5)
Answer: The two 4-tuples are not equal because their second and third components are in different positions.

Why It Matters

Vectors in Rn\mathbb{R}^n, coordinate points, and database records are all modeled as n-tuples. Understanding them is essential for linear algebra courses and for working with Cartesian products in discrete math. In computer science, function arguments and data structures like Python tuples rely on the same ordered-sequence concept.

Common Mistakes

Mistake: Treating an n-tuple like a set, assuming order doesn't matter or that duplicates collapse.
Correction: In a tuple, order is significant and repeated elements are counted separately. (2,3)(2, 3) and (3,2)(3, 2) are different tuples, and (1,1)(1, 1) is not the same as (1)(1).