Let expression

Key takeaways

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

Reference excerpt

In computer science, a "let" expression associates a function definition with a restricted scope. The "let" expression may also be defined in mathematics, where it associates a Boolean condition with a restricted scope. The "let" expression may be considered as a lambda abstraction applied to a value. Within mathematics, a let expression may also be considered as a conjunction of expressions, within an existential quantifier which restricts the scope of the variable. The let expression is present in many functional languages to allow the local definition of expression, for use in defining another expression. The let-expression is present in some functional languages in two forms; let or "let rec". Let rec is an extension of the simple let expression which uses the fixed-point combinator to implement recursion.

History Dana Scott's LCF language was a stage in the evolution of lambda calculus into modern functional languages. This language introduced the let expression, which has appeared in most functional languages since that time. The languages Scheme, ML, and more recently Haskell have inherited let expressions from LCF. Stateful imperative languages such as ALGOL and Pascal essentially implement a let expression, to implement restricted scope of functions, in block structures. A closely related "where" clause, together with its recursive variant "where rec", appeared already in Peter Landin's The mechanical evaluation of expressions.

Description A "let" expression defines a function or value for use in another expression. As well as being a construct used in many functional programming languages, it is a natural language construct often used in mathematical texts. It is an alternate syntactical construct for a where clause.

In both cases the whole construct is an expression whose value is 5. Like the if-then-else the type returned by the expression is not necessarily Boolean. A let expression comes in 4 main forms,

In functional languages the let expression defines functions which may be called in the expression. The scope of the function name is limited to the let expression structure. In mathematics, the let expression defines a condition, which is a constraint on the expression. The syntax may also support the declaration of existentially quantified variables local to the let expression. The terminology, syntax and semantics vary from language to language. In Scheme, let is used for the simple form and let rec for the recursive form. In ML let marks only the start of a block of declarations with fun marking the start of the function definition. In Haskell, let may be mutually recursive, with the compiler figuring out what is needed.

Definition A lambda abstraction represents a function without a name. This is a source of the inconsistency in the definition of a lambda abstraction. However lambda abstractions may be composed to represent a function with a name. In this form the inconsistency is removed. The lambda term,

( λ f . z ) ( λ x . y ) {\displaystyle (\lambda f.z)\ (\lambda x.y)}

is equivalent to defining the function f {\displaystyle f} by f x = y {\displaystyle f\ x=y} in the expression z {\displaystyle z} , which may be written as the let expression;

let ⁡ f x = y in ⁡ z {\displaystyle \operatorname {let} f\ x=y\operatorname {in} z}

The let expression is understandable as a natural language expression. The let expression represents the substitution of a variable for a value. The substitution rule describes the implications of equality as substitution.

Let definition in mathematics In mathematics the let expression is described as the conjunction of expressions. In functional languages the let expression is also used to limit scope. In mathematics scope is described by quantifiers. The let expression is a conjunction within an existential quantifier.

( ∃ x E ∧ F ) ⟺ let ⁡ x : E in ⁡ F {\displaystyle (\exists xE\land F)\iff \operatorname {let} x:E\operatorname {in} F}

where E and F are of type Boolean. The let expression allows the substitution to be applied to another expression. This substitution may be applied within a restricted scope, to a sub expression. The natural use of the let expression is in application to a restricted scope (called lambda dropping). These rules define how the scope may be restricted;

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Let expression

Start with the simplest possible case. Write down what Let expression claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Let expression 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 Let expression 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 Let expression

In research: Let expression appears in mathematics 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 Let expression 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: Let expression is common in secondary-school and first-year university syllabi. It links to neighbouring topics Lambda calculus, so understanding it makes those chapters shorter.
In everyday life: Look for Let expression 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 Let expression in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Let expression 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 Let expression out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Let expression in simple terms?

In computer science, a "let" expression associates a function definition with a restricted scope. The "let" expression may also be defined in mathematics, where it associates a Boolean condition with a restricted scope.

Why does Let expression matter?

Because it connects several mathematics 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 Let expression?

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 Let expression.