ordered pair
C2Formal, Technical, Academic
Definition
Meaning
In mathematics and logic, a pair of objects where the order of the objects is significant; the first and second elements must be distinguished.
Can be extended metaphorically to any situation where two components are considered together and their sequence or designated roles (first, second) are crucial to the concept.
Linguistics
Semantic Notes
The concept is foundational in set theory, where it is often defined as (a, b) = {{a}, {a, b}}. Its meaning is almost exclusively technical and non-negotiable in that context.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or orthographic differences; the term is identical in both varieties. Usage is confined to technical mathematical contexts.
Connotations
Purely technical, with no cultural or regional connotations.
Frequency
Extremely low frequency in general language; frequency spikes only in mathematical, computer science, and formal logic texts. No regional variation in frequency.
Vocabulary
Collocations
Grammar
Valency Patterns
[ordered pair] of [N]the [ordered pair] ([element1], [element2])Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “None”
Usage
Context Usage
Business
Virtually never used.
Academic
Core term in mathematics, computer science, formal logic, and related disciplines.
Everyday
Extremely rare, only in highly specific discussions (e.g., explaining map coordinates).
Technical
The primary domain. Used to define relations, functions, and coordinate systems.
Examples
By Part of Speech
verb
British English
- The function is defined by pairing and ordering the elements.
American English
- We need to pair and order the inputs to create the correct mapping.
adverb
British English
- The elements were listed orderedly, as a pair.
American English
- The data is structured ordered-pair-wise.
adjective
British English
- The ordered-pair representation is fundamental.
American English
- They used an ordered-pair notation for the coordinates.
Examples
By CEFR Level
- On a map, (3, 5) is a point. The first number is left-right, the second is up-down.
- In maths, (5, 7) and (7, 5) are different ordered pairs because the order changes.
- A relation is simply a set of ordered pairs, linking elements from a domain to a codomain.
- The Kuratowski definition of the ordered pair (a, b) as {{a}, {a, b}} ensures that (a, b) = (c, d) iff a = c and b = d.
Learning
Memory Aids
Mnemonic
Think of a cinema ticket: (Row F, Seat 12) is not the same as (Seat 12, Row F). The ORDER in the pair matters.
Conceptual Metaphor
A precise, labelled container with two slots: Slot 1 and Slot 2.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- The Russian term "упорядоченная пара" is a direct and accurate translation. No trap beyond the highly technical nature of the concept.
Common Mistakes
- Using it interchangeably with 'set' or 'unordered pair'.
- Omitting the parentheses or commas in notation.
- Reversing the elements without understanding the consequence.
Practice
Quiz
Which of the following best describes an 'ordered pair'?
FAQ
Frequently Asked Questions
No. In an ordered pair, (2, 3) ≠ (3, 2) unless 2 = 3. The order is part of the identity of the pair.
A set {a, b} is unordered: {a, b} = {b, a}. An ordered pair (a, b) distinguishes the first element from the second.
It is the foundational building block for defining Cartesian products, relations, functions, and coordinate systems in mathematics and computer science.
Yes. It is a perfectly valid ordered pair where the first and second elements are equal.