Product type

Key takeaways

Product type belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Product type to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Product type from memory before moving on to harder problems.

Reference excerpt

In programming languages and type theory, a product of types is another, compounded, type in a structure. The "operands" of the product are types, and the structure of a product type is determined by the fixed order of the operands in the product. An instance of a product type retains the fixed order, but otherwise may contain all possible instances of its primitive data types. The expression of an instance of a product type will be a tuple, and is called a "tuple type" of expression. A product of types is a direct product of two or more types. If there are only two component types, it can be called a "pair type". For example, if two component types A {\displaystyle A} and B {\displaystyle B} are the set of all possible values of that type, the product type written A × B {\displaystyle A\times B} contains elements that are pairs ( a , b ) {\displaystyle (a,b)} , where a {\displaystyle a} and b {\displaystyle b} are instances of A {\displaystyle A} and B {\displaystyle B} respectively. The pair type is a special case of the dependent pair type, where the type B {\displaystyle B} may depend on the instance picked from A {\displaystyle A} . In many languages, product types take the form of a record type, for which the components of a tuple can be accessed by label. In languages that have algebraic data types, as in most functional programming languages, algebraic data types with one constructor are isomorphic to a product type. In the Curry–Howard correspondence, product types are associated with logical conjunction (AND) in logic. The notion directly extends to the product of an arbitrary finite number of types (an n {\displaystyle n} -ary product type), and in this case, it characterizes the expressions that behave as tuples of expressions of the corresponding types. A degenerate form of product type is the unit type: it is the product of no types. In call-by-value programming languages, a product type can be interpreted as a set of pairs whose first component is a value in the first type and whose second component is a value in the second type. In short, it is a cartesian product and it corresponds to a product in the category of types. Most functional programming languages have a primitive notion of product type. For instance, the product T 1 × T 2 × . . . × T n {\displaystyle T_{1}\times T_{2}\times ...\times T_{n}} is written T1 * T2 * ... * Tn in ML and (T1, T2, ..., Tn) in Haskell. In both these languages, tuples are written (v1, v2, ..., vn) and the components of a tuple are extracted by pattern-matching. Additionally, many functional programming languages provide more general algebraic data types, which extend both product and sum types. Product types are the dual of sum types.

Product types in programming languages C++ defines the class std::tuple (expressed tuple<Ts...> using variadic templates), and for the specific case of two elements defines std::pair (expressed pair<T, U>). std::tuple can be empty (tuple<>). C#/.NET Framework defines the class System.Tuple. There are specific instantiations for 1 to 8 elements. For the specific case of two elements (a pair), it uses Tuple<T1, T2>. In order to create a tuple with nine or more components, the final parameter TRest of Tuple<T1, T2, T3, T4, T5, T6, T7, TRest> is supplied as another tuple. For iterating over collections like dictionary types, the class System.Collections.Generic.KeyValuePair (expressed KeyValuePair<TKey, TValue>) is provided. Go does not have a tuple type, but can express multiple return values in a function as a sort of tuple. Haskell has a data type Data.Tuple. Java does not have a general tuple type, but JavaFX has a type javafx.util.Pair (expressed Pair<K, V>). For iterating over associative containers such as java.util.Map, a pair in the map is expressed as Map.Entry<K, V>. Kotlin does not have a general tuple type, but has classes kotlin.Pair (expressed Pair<A, B>) and kotlin.Triple (expressed Triple<A, B, C>). Python has a tuple collection which can be annotated as typing.Tuple (expressed Tuple[T1, T2, ..., TN]). Rust defines the primitive tuple type, expressed as (T1, T2, ..., TN), and a pair is just (T, U). Scala defines the class scala.Tuple, which supports between 2 and 22 objects as scala.Tuple2 (expressed as Tuple2[A, B]) to scala.Tuple22 (expressed as Tuple22[A, B, ..., V]). Swift expresses tuples as (T1, T2, ..., TN).

See also Product (category theory) Cartesian product Record (computer science) Struct (C programming language) Sum type Quotient type

References product type at the nLab Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study. See section 1.5.

Worked examples

Example 1 — a first encounter with Product type

Start with the simplest possible case. Write down what Product type claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Product type before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Product type ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Product type

In research: Product type appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Product type in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Product type is common in secondary-school and first-year university syllabi. It links to neighbouring topics Composite data types, Data types, Type theory, so understanding it makes those chapters shorter.
In everyday life: Look for Product type outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Product type in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Product type means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Product type out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Product type in simple terms?

In programming languages and type theory, a product of types is another, compounded, type in a structure. The "operands" of the product are types, and the structure of a product type is determined by the fixed order of the operands in the product.

Why does Product type matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Product type?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Product type.