# Is there a good word for a square-rectangle relationship?

Any given square is always a rectangle, but a rectangle isn't necessarily a square, so squares and rectangles have a _ relationship. I've been noticing this sort of thing everywhere ever since I noticed that I didn't know a good word for it. I've been calling it a container relationship because one class of things is contained within a larger class of things. However, a "container" relationship doesn't sound very good and doesn't really convey the meaning on its own. A more elegant word would be helpful.

Edit: A specific definition would be something like: a is always b. b is sometimes a.

• @DanBron I like your answer better than mine, but yes, there is a mathematical/logical word, it is "subset" (intransitive) and "set" being the outer (abstraction layer)
– Mike
Sep 15, 2014 at 20:18
• In Object Oriented programming this is an "is a" relationship. Sep 15, 2014 at 21:12
• If you're coming from programming you should note that squares are not rectangles. /cc @MartinSmith
– user36720
Sep 15, 2014 at 23:18
• @djechlin In programming, whether an "is a" relationship between squares and rectangles is appropriate, depends on the design. Saying it's appropriate without further info is wrong. Saying it's inappropriate without further info is also wrong.
– hvd
Sep 16, 2014 at 5:59
• I've managed for 65 years now with "sorta kinda like". Sep 16, 2014 at 20:28

You can utilize the word "subset" for this usage.

Squares are a "subset" of Rectangles.

Meaning, they are within the "set" of Rectangles, but not all rectangles are squares.

sub·set ˈsəbˌset
noun noun: subset;
plural noun: subsets;
noun: sub-set;
plural noun: sub-sets
DEFINTION

a part of a larger group of related things.

synonyms: subcategory, branch, subdivision, subsection, subsidiary

Examples:

"the quartet is a subset of our orchestral group"

Usage in MATHEMATICS

a set of which all the elements are contained in another set.

• Google Define is not a citable reference, because it gives different results to different people at different times, and sometimes even to the same people at different times. You must cite an actual dictionary.
– tchrist
Sep 15, 2014 at 20:30
• Subset is definitely the right word to use to describe the square, and I would call the rectangle a superset; but the real question here is whether there is a word for the relationship between a subset and a superset. You wouldn't say that square and rectangle have a subset relationship (or, pace Dan's suggestion, a subsidiary relationship), so as a strict answer to the question, I fear it doesn't work. Sep 15, 2014 at 20:37
• A word for the relationship between a superset and a subset is hierarchical. Sep 15, 2014 at 20:42
• It can also be helpful to realize that rectangles are a generalization of squares. Rectangles are what you get if you keep the restriction of right angles, but relax the restriction of equal length sides. The rhombus is another generalization when you relax the angles but not the lengths. There are lots of possibilities. It's good to note that rectangle isn't the only superset of square, and that not every pair of supersets of square have a subclass/superclass relationship with each other. Sep 16, 2014 at 3:36
• @JoshuaTaylor Also worth noting that the converse of the generalization relationship is that squares are a special case of rectangles. Sep 16, 2014 at 4:15

Square is a hyponym of rectangle, which is a hypernym of square. The wikipedia article Hyponymy and hypernymy says

In linguistics, a hyponym is a word or phrase whose semantic field is included within that of another word, its hypernym (sometimes spelled hyperonym outside of the natural language processing community). In simpler terms, a hyponym shares a type-of relationship with its hypernym. For example, "pigeon", "crow", "eagle" and "seagull" are all hyponyms of "bird" (their hypernym); which, in turn, is a hyponym of "animal".

Edit: For more precision, one should add qualifying phrases like “the word” or “the set”:

• the word square is a hyponym of the word rectangle
• the set of squares is a subset of the set of rectangles

However, I think subclass [a term mentioned in Joshua Taylor's comments] can be used without added qualifiers.Wiktionary gives the following definition of subclass relevant within computing: “In object-oriented programming, an object class derived from another class (its superclass) from which it inherits a base set of properties and methods”.

• squares are a subclass of rectangles

Speaking mathematically, a class is a less-determinate category or collection of things than is a set. From en.wiktionary, class means “A group, collection, category or set sharing characteristics or attributes”, and subclass means “A rank directly below class”.

• I'm uneasy with this usage (I was hoping nobody would suggest it). Knife and cutlery, chair and furniture, sock and clothing, certainly: but set terminology seems far more appropriate for use in the mathematical field. Terms are almost always well-defined in maths, whereas one person's 'hurricane' may be a hyponym of another person's. And many public squares aren't even rectangular. Sep 16, 2014 at 6:55
• This is the English Language & Usage group, and these terms are about sets of semantic meaning, and as such are also mathematically appropriate! Sep 16, 2014 at 12:07
• I wouldn't ever recommend those words, the will just be perceived as jargon... and they are phonically close with like every pair of hyper/hypo words... Sep 16, 2014 at 13:49
• These are the perfect terms to describe the relationship between the words square and rectangle, though perhaps not for the relationship between squares and rectangles themselves. Sep 17, 2014 at 14:48
• @Grady I think saying every pair is a bit of hypobole. I mean... crap... Sep 17, 2014 at 16:26

A square is a special type, or a specialisation of rectangle.

A rectangle is a more general type, a generalisation of a square.

They are in a hierarchical relationship. They are in a specialisation relationship.

The relationship between a square and a rectangle is "type of". A square is a type of rectangle, but a rectangle is not a type of square.

I'm not aware of a single word that means "type of".

In engineering and programming circles, this relationship is also described as "is a". Another similar relationship in this context is are "has a".

A rectangle definitely does not "contain" a square, in the sense of the original question: that would be the wrong thing to say to mean that a square is a type of rectangle. When you say "a rectangle contains a square" you mean "has a" not "is a".

It's also "obscure" to think of a "square" as a subset of a "rectangle". The set of squares might be a subset of the set of rectangles, but "square" and "rectangle" in this context are types (of shapes), not sets.

• The question didn't claim that a rectangle contains a square, only that the class of rectangles contains the class of squares, which is true. Sep 17, 2014 at 4:40
• I was addressing the questions use of the term "container". My point is that the relationship between a rectangle and a square is not about "containment" - it is about "type of". It surely is the case the the set of all possible rectangles contains the set off all possible squares, but (despite the most popular answer!) it is not the case that the relationship the question is looking for is one of "subset". A single square is a "type of "rectangle", completely independent from the existence of or consideration of "sets". Sep 17, 2014 at 12:19

You can consider the set of all squares and the set of all rectangles and how they overlap with one another. In mathematical jargon we would say Squares are a proper subset of rectangles. or Rectangles are a proper superset of squares.

In normal, non-technical English, Squares are a subset of rectangles. will generally be understood to mean this.

You can also consider the meanings of square and rectangle and how they relate to one another. The meaning of square can be expressed in terms of being a rectangle with additional restriction: A square is an equilateral rectangle. So the relationship could be expressed as: A square is a kind of rectangle. There are a wide range of other ways of expressing kind of. Depending on the direction, the relationship itself is called specialization or generalization.

Most people are more comfortable expressing things in terms of subsets or specialization than they are in terms of supersets or generalization. So unless you are in a technical context, or the direction is very important to your meaning, it's best to stick to expressing these relationships in terms of specialization.

• Do not forget that Squares are also a subset of Rhombi. Sep 17, 2014 at 18:43
• @Victor It's not particularly relevant to the question. Sep 17, 2014 at 19:48

The relation is called inclusion. A set includes all of its subsets. The set of rectangles includes the set of squares.

Expressed in terms of membership, each member of a subset of a set S is also a member of S. Each square is a rectangle.

(And a set includes itself as a subset. A proper subset of a set S is a subset that is not the same as S.)

Expressed in terms of predicates, the relation is implication. Being a square implies being a rectangle; squareness implies rectangleness.

A square is special case of rectangles.

As Mike said, subset is correct. In my opinion, by saying special case the uniqueness of squares amongst rectangles can be emphasised.

They share a "specialization" relationship if you want to use a software term.

I had the same original question as you. Perhaps "mutual inclusion" or "mutually inclusive." I wad having an argument with someone when I said I liked to ride motorcycles because they are fun, and she replied "oh, it's a game to you?" Games are fun, leading her to this conclusion. But, not all fun derives from games. Fun can come from other forms. I thought about the rectangle/square relation, and it amazed me that I didn't know the term for their relationship. After reading the responses here, saying that two things are not mutually inclusive seems like a reasonable phrase.

JH