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$A$: domain (set of possible inputs)
$B$: co-domain (set of allowed outputs)

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$A$: domain (set of possible inputs)
$B$: co-domain (set of allowed outputs)

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$A$: co-domain (set of allowed outputs)
$B$: domain (set of possible inputs)

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$A$: range (set of actual outputs)
$B$: domain (set of possible inputs)

","correct":false,"explanation":"This confuses the range with the domain and reverses the mapping. $A$ is the domain of inputs, while $B$ is the co-domain containing all allowed outputs."},{"text":"

$A$: domain (set of possible inputs)
$B$: range (set of actual outputs)

","correct":false,"explanation":"The target set $B$ is the co-domain, not necessarily the range. The range contains only outputs actually produced and may be a proper subset of $B$."}]},{"id":"b67b4693-9496-4d2d-a7d3-de5867ce0abd","cardType":"cloze","points":null,"front":"In functional programming, a function does not have to use every value in its [ ].","back":"In functional programming, a function does not have to use every value in its co-domain.","mcqOptions":[{"text":"co-domain","correct":true,"explanation":"The co-domain contains every output value that the function is allowed to produce. Its actual outputs form the range, which may contain only some co-domain values."},{"text":"domain","correct":false,"explanation":"The domain is the set of permitted inputs, and a total function must be defined for every member of it. The function need not produce every value in its co-domain."},{"text":"range","correct":false,"explanation":"The range is defined as the set of values the function actually produces, so every range value is used. Unused allowed outputs belong to the co-domain instead."},{"text":"input set","correct":false,"explanation":"The input set is another description of the domain, whose values must have defined mappings for a total function. The set that may contain unused values is the co-domain."}]},{"id":"b82ee884-d091-410f-add1-4eb04f489a0b","cardType":"basic","points":3,"front":"What are the four properties that make a value a first-class object?","back":"

Can be assigned to a variable
Can be passed as an argument
Can be returned from a function
Can appear in expressions

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Can be assigned to a variable
Can be passed as an argument
Can be returned from a function
Can appear in expressions

","correct":true,"explanation":"A first-class object can be manipulated like any ordinary value: stored in a variable, supplied to a function, returned as a result, and used within expressions."},{"text":"

Can be assigned to a variable
Can be passed as an argument
Can be returned from a function
Can be executed at compile time

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Can be assigned to a variable
Can modify global state
Can be returned from a function
Can appear in expressions

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Can be given a unique name
Can be passed as an argument
Can be returned from a function
Can appear in expressions

","correct":false,"explanation":"A first-class value need not have a unique name, as anonymous functions demonstrate. It must instead be assignable to a variable, as well as passed, returned, and used in expressions."}]},{"id":"329daad6-59b5-4193-b4ef-1acc348fb016","cardType":"cloze","points":null,"front":"A [ ] is a function that takes another function as an argument or returns a function as a result.","back":"A higher-order function is a function that takes another function as an argument or returns a function as a result.","mcqOptions":[{"text":"higher-order function","correct":true,"explanation":"A higher-order function treats functions as first-class values by accepting a function as an argument, returning a function as its result, or doing both."},{"text":"pure function","correct":false,"explanation":"A pure function is characterised by deterministic output and no side effects, not by accepting or returning functions. The required term is higher-order function."},{"text":"recursive function","correct":false,"explanation":"A recursive function calls itself directly or indirectly. A function that instead accepts another function or returns one is called a higher-order function."},{"text":"anonymous function","correct":false,"explanation":"An anonymous function is defined without a name, but it need not accept or return another function. That behaviour defines a higher-order function."}]},{"id":"70b94541-d2bd-48dd-a040-88faa563c4c7","cardType":"basic","points":2,"front":"How is an imperative programming language defined in contrast to functional programming?","back":"Programs consist of sequences of commands that update state.","mcqOptions":[{"text":"Programs consist of sequences of commands that update state.","correct":true,"explanation":"Imperative programs specify a sequence of instructions that alter stored program state. This contrasts with functional programs, which primarily evaluate expressions and avoid mutable state."},{"text":"Programs consist of nested expressions that avoid state.","correct":false,"explanation":"This describes the functional approach rather than the imperative one. Imperative programs execute sequences of commands that update the program’s state over time."},{"text":"Programs consist of classes and objects that exchange messages.","correct":false,"explanation":"Classes, objects, and message passing characterise object-oriented programming. Imperative programming is defined by executing sequences of commands that update program state."},{"text":"Programs consist of facts and rules that support inference.","correct":false,"explanation":"Facts, rules, and inference characterise logic programming. An imperative language instead expresses programs as ordered sequences of commands that update state."}]},{"id":"4aa657bc-189d-4175-9a9a-f5f5cdd47505","cardType":"basic","points":1,"front":"What is function application?","back":"Giving specific argument values to a function to evaluate it.","mcqOptions":[{"text":"Giving specific argument values to a function to evaluate it.","correct":true,"explanation":"Function application supplies particular arguments to a function so that its expression can be evaluated and a corresponding result can be produced."},{"text":"Giving a specific function name to a definition to store it.","correct":false,"explanation":"Naming and storing a definition creates or binds a function but does not apply it. Function application means supplying specific argument values for evaluation."},{"text":"Giving specific return values to a function to define it.","correct":false,"explanation":"Return values are produced by evaluating a function rather than supplied beforehand. Function application supplies argument values to the function so it can be evaluated."},{"text":"Giving another function definition to a function to compose it.","correct":false,"explanation":"Combining functions describes function composition, although functions may also be arguments. Function application more generally means giving specific argument values to a function for evaluation."}]},{"id":"bee7831f-8d22-497b-bc82-52bc4f2e771e","cardType":"basic","points":1,"front":"How is a two-argument function written using Cartesian product notation?","back":"$$f: A \\times B \\rightarrow C$","mcqOptions":[{"text":"$$f: A \\times B \\rightarrow C$","correct":true,"explanation":"The Cartesian product $A \\times B$ is the set of ordered pairs containing one value from each input set. The function maps each pair to an element of $C$."},{"text":"$$f: A \\rightarrow B \\times C$","correct":false,"explanation":"This notation describes one input from $A$ producing an ordered-pair output in $B \\times C$. Two arguments using Cartesian product notation require the domain $A \\times B$."},{"text":"$$f: A \\cup B \\rightarrow C$","correct":false,"explanation":"A union allows a single input drawn from either $A$ or $B$; it does not combine two arguments. The required domain is the Cartesian product $A \\times B$."},{"text":"$$f: A \\rightarrow B \\rightarrow C$","correct":false,"explanation":"This is curried notation, where one argument produces a function that accepts the next. Cartesian product notation instead writes the paired input domain as $A \\times B$."}]},{"id":"1841c509-9f23-4ed9-84a1-4a1f5f29c3e7","cardType":"cloze","points":null,"front":"In the notation $\\text{add}: \\text{integer} \\times \\text{integer} \\rightarrow \\text{integer}$, the input is formally treated as a single [ ].","back":"In the notation $\\text{add}: \\text{integer} \\times \\text{integer} \\rightarrow \\text{integer}$, the input is formally treated as a single ordered pair.","mcqOptions":[{"text":"ordered pair","correct":true,"explanation":"An element of $\\text{integer} \\times \\text{integer}$ is an ordered pair of integers. The order matters because the first and second argument can play different roles."},{"text":"unordered pair","correct":false,"explanation":"Cartesian products contain ordered pairs, not unordered pairs. Swapping the two integers may change the function’s result, so their first and second positions must be preserved."},{"text":"Cartesian product","correct":false,"explanation":"The Cartesian product is the complete set of possible ordered pairs, not one individual input. A particular function input is a single ordered pair from that set."},{"text":"scalar value","correct":false,"explanation":"A scalar is a single atomic quantity and does not preserve two argument positions. Here the single formal input is an ordered pair containing two integers."}]},{"id":"37947405-8f53-406b-a61c-9b2f225106e6","cardType":"basic","points":2,"front":"What is partial function application?","back":"Applying a function to fewer arguments than it requires to produce a new function.","mcqOptions":[{"text":"Applying a function to fewer arguments than it requires to produce a new function.","correct":true,"explanation":"Partial function application supplies fewer arguments than the function requires, leaving a new function that waits for the remaining arguments."},{"text":"Applying a function to all its arguments to produce a new function.","correct":false,"explanation":"Applying all required arguments normally produces the function’s result, not a new partially applied function. Partial application applies fewer arguments and returns a new function."},{"text":"Applying a function to fewer arguments than it requires to produce a single result.","correct":false,"explanation":"Supplying fewer arguments cannot normally produce the final result because some inputs are still missing. It produces a new function that accepts the remaining arguments."},{"text":"Applying a function to more arguments than it requires to produce a new function.","correct":false,"explanation":"Providing more arguments than a function requires is an argument error, not partial application. The correct process applies fewer arguments and returns a new function."}]},{"id":"f9ed2fe9-0d1d-4003-a772-8ff7d8a1972c","cardType":"cloze","points":null,"front":"Partial function application is only possible because functions are treated as [ ].","back":"Partial function application is only possible because functions are treated as first-class objects.","mcqOptions":[{"text":"first-class objects","correct":true,"explanation":"Functions are first-class objects, so they can be stored, passed as arguments and returned as results. This allows a partially applied function to be produced and used later."},{"text":"first-order values","correct":false,"explanation":"First-order values cannot themselves be passed around as functions, so they do not support partial application. The necessary feature is treating functions as first-class objects."},{"text":"compiled procedures","correct":false,"explanation":"Compilation describes how code is translated or executed, not whether functions can be manipulated as values. Partial application requires functions to be treated as first-class objects."},{"text":"constant expressions","correct":false,"explanation":"A constant expression has a fixed value and does not explain how a function can accept arguments later. Partial application depends on functions being treated as first-class objects."}]},{"id":"87e213bf-00f8-487f-8afc-cfbf327a724c","cardType":"basic","points":2,"front":"Why is the function type $A \\rightarrow B \\rightarrow C$ described as right-associative?","back":"It is interpreted as $A \\rightarrow (B \\rightarrow C)$, meaning it takes $A$ and returns a function of type $B \\rightarrow C$.","mcqOptions":[{"text":"It is interpreted as $A \\rightarrow (B \\rightarrow C)$, meaning it takes $A$ and returns a function of type $B \\rightarrow C$.","correct":true,"explanation":"Function arrows associate from the right, so $A \\rightarrow B \\rightarrow C$ is grouped as $A \\rightarrow (B \\rightarrow C)$. After receiving an $A$, it returns a function awaiting a $B$."},{"text":"It is interpreted as $(A \\rightarrow B) \\rightarrow C$, meaning it takes a function from $A$ to $B$ and returns $C$.","correct":false,"explanation":"This groups the first arrow to the left, which is left-associative rather than right-associative. The correct interpretation is $A \\rightarrow (B \\rightarrow C)$."},{"text":"It is interpreted as $A \\rightarrow (B \\rightarrow C)$, meaning it takes $B$ and returns a function of type $A \\rightarrow C$.","correct":false,"explanation":"Although the parentheses are correct, the input order has been reversed. The type first takes $A$, then returns a function that takes $B$ to produce $C$."},{"text":"It is interpreted as $(A \\rightarrow B) \\rightarrow C$, meaning it takes $C$ and returns a function of type $A \\rightarrow B$.","correct":false,"explanation":"Both the grouping and the direction are wrong here. Right-associativity gives $A \\rightarrow (B \\rightarrow C)$, so the type takes $A$ and returns a $B \\rightarrow C$ function."}]},{"id":"45024203-c4fa-4715-9cb4-8133b135db34","cardType":"basic","points":1,"front":"If $h: A \\rightarrow B \\rightarrow C \\rightarrow D$, what is the type of the result after applying the first two arguments?","back":"$$C \\rightarrow D$","mcqOptions":[{"text":"$$C \\rightarrow D$","correct":true,"explanation":"The first application consumes the $A$ argument and the second consumes the $B$ argument. The function still expects a $C$ and then produces a $D$, giving $C \\rightarrow D$."},{"text":"$$A \\rightarrow B$","correct":false,"explanation":"Applying only the first two arguments consumes the $A$ and $B$ inputs, so the remaining result cannot still begin with $A \\rightarrow B$. The remaining type is $C \\rightarrow D$."},{"text":"$$B \\rightarrow C$","correct":false,"explanation":"This reflects removing only one argument or misreading the chain of arrows. After applying arguments of types $A$ and $B$, the remaining function takes $C$ and returns $D$."},{"text":"$$D \\rightarrow C$","correct":false,"explanation":"The final input-output relationship has been reversed. The remaining function accepts $C$ and returns $D$, so its type is $C \\rightarrow D$, not $D \\rightarrow C$."}]},{"id":"29d1c1f9-f37b-4edb-a04e-970f6b2f9bbb","cardType":"basic","points":1,"front":"What is produced by function composition?","back":"A new function created by combining two existing functions.","mcqOptions":[{"text":"A new function created by combining two existing functions.","correct":true,"explanation":"Function composition combines two existing functions so that the output of one becomes the input of the other, producing a new function."},{"text":"A new value created by combining two existing functions.","correct":false,"explanation":"Composition combines functions to create another function, not an immediately computed value. The result remains applicable to an input and is a new function."},{"text":"A new function created by duplicating one existing function.","correct":false,"explanation":"Duplicating a function does not combine the behaviours of two functions and is not composition. Composition uses two existing functions to create a new function."},{"text":"A new function created by replacing two existing functions.","correct":false,"explanation":"Composition does not replace the original functions; it combines their operations into another callable function. The existing functions remain available and a new function is produced."}]},{"id":"dcf6d4e6-4e71-476e-a658-8db3ef8e13c8","cardType":"cloze","points":null,"front":"The composition $g \\circ f$ means that $\\text{[...]}$ is applied first, then $g$ is applied to the result.","back":"The composition $g \\circ f$ means that $\\boxed{f}$ is applied first, then $g$ is applied to the result.","mcqOptions":[{"text":"$$f$","correct":true,"explanation":"In $g \\circ f$, function composition applies the right-hand function first. Therefore $f$ receives the original input, and $g$ is then applied to the result."},{"text":"$$g$","correct":false,"explanation":"This reverses the order suggested by the notation. In $g \\circ f$, the right-hand function $f$ is applied first, followed by $g$."},{"text":"$$f \\circ g$","correct":false,"explanation":"This is a composition expression rather than the function applied first, and it reverses the order being described. For $g \\circ f$, $f$ is applied first."},{"text":"$$g \\circ f$","correct":false,"explanation":"The whole composition is not the first individual function applied. The rightmost component, $f$, acts on the input first, after which $g$ processes its result."}]},{"id":"30468a32-8e8f-43d3-92f2-fad6eba2b633","cardType":"cloze","points":null,"front":"The composition $g \\circ f$ means that $f$ is applied first, then $\\text{[...]}$ is applied to the result.","back":"The composition $g \\circ f$ means that $f$ is applied first, then $\\boxed{g}$ is applied to the result.","mcqOptions":[{"text":"$$g$","correct":true,"explanation":"Composition applies functions from right to left. Thus $f$ processes the input first, and $g$ is then applied to the value returned by $f$."},{"text":"$$f$","correct":false,"explanation":"This repeats the first function instead of identifying the second stage. In $g \\circ f$, $f$ is applied first and then $g$ is applied to its result."},{"text":"$$f \\circ g$","correct":false,"explanation":"This denotes the reversed composition and does not name the function applied second in $g \\circ f$. After $f$ runs, the result is passed to $g$."},{"text":"$$g \\circ f$","correct":false,"explanation":"The complete composition is not the function applied to the intermediate result. The second individual function is $g$, which receives the output from $f$."}]},{"id":"200dc07b-25d6-4054-9a47-0df372b0c253","cardType":"basic","points":1,"front":"What is the mathematical definition of $(g \\circ f)(x)$?","back":"$$g(f(x))$","mcqOptions":[{"text":"$$g(f(x))$","correct":true,"explanation":"The rightmost function in a composition is applied first. Therefore $f$ receives $x$, and $g$ is applied to the resulting value, giving $g(f(x))$."},{"text":"$$f(g(x))$","correct":false,"explanation":"This applies $g$ first and then $f$, reversing the order of $g \\circ f$. The definition is $g(f(x))$, where $f$ acts on $x$ before $g$."},{"text":"$$g(x)f(x)$","correct":false,"explanation":"This multiplies the two function outputs rather than feeding one output into the other. Composition is written $g(f(x))$, showing that $f(x)$ becomes the input to $g$."},{"text":"$$g(x+f(x))$","correct":false,"explanation":"This adds the original input to the output of $f$, which is not part of composition. The input to $g$ is simply $f(x)$, so the result is $g(f(x))$."}]},{"id":"06ec90f5-bc20-45ac-88c9-bf0775b93db5","cardType":"basic","points":2,"front":"For the composition $g \\circ f$ to be valid, what must be true about the types of $f$ and $g$?","back":"The output type (co-domain) of $f$ must match the input type (domain) of $g$.","mcqOptions":[{"text":"The output type (co-domain) of $f$ must match the input type (domain) of $g$.","correct":true,"explanation":"Composition applies $f$ first and then passes its result into $g$, so every value produced by $f$ must have the type accepted by $g$."},{"text":"The input type (domain) of $f$ must match the output type (co-domain) of $g$.","correct":false,"explanation":"This reverses the required compatibility by treating $g$ as though it were applied first. In $g \\circ f$, the output type of $f$ must match the input type of $g$."},{"text":"The input type (domain) of $f$ must match the input type (domain) of $g$.","correct":false,"explanation":"Matching the two input types does not ensure that $g$ can receive the result of $f$. The output type of $f$ must match the input type of $g$."},{"text":"The output type (co-domain) of $f$ must match the output type (co-domain) of $g$.","correct":false,"explanation":"Matching the output types is unnecessary and does not make the intermediate value valid. The output type of $f$ must instead match the input type of $g$."}]},{"id":"0aaecaec-b588-434a-a989-3830718bab1a","cardType":"cloze","points":null,"front":"If $f: A \\rightarrow B$ and $g: B \\rightarrow C$, the type of $g \\circ f$ is $\\text{[...]}$.","back":"If $f: A \\rightarrow B$ and $g: B \\rightarrow C$, the type of $g \\circ f$ is $\\boxed{A \\rightarrow C}$.","mcqOptions":[{"text":"$$A \\rightarrow C$","correct":true,"explanation":"The composition first maps an input from $A$ through $f$ to $B$, then through $g$ to $C$, giving the overall type $A \\rightarrow C$."},{"text":"$$B \\rightarrow C$","correct":false,"explanation":"This is the type of $g$, not the complete composition. Since $g \\circ f$ begins with an input to $f$, its domain is $A$ and its co-domain is $C$."},{"text":"$$A \\rightarrow B$","correct":false,"explanation":"This is the type of $f$ alone and ignores the subsequent application of $g$. The composition starts in $A$ but finishes in $C$, so its type is $A \\rightarrow C$."},{"text":"$$C \\rightarrow A$","correct":false,"explanation":"This reverses the direction of the composite mapping. The function $f$ is applied before $g$, so $g \\circ f$ maps from $A$ to $C$, not from $C$ to $A$."}]},{"id":"a7f7f62e-8b7d-40b4-9cd7-916579412937","cardType":"basic","points":1,"front":"In functional programming, what is the domain?","back":"The set from which input values are chosen.","mcqOptions":[{"text":"The set from which input values are chosen.","correct":true,"explanation":"A function’s domain specifies the set of values that may be supplied as its inputs, distinguishing it from the co-domain, which concerns possible outputs."},{"text":"The set from which output values are chosen.","correct":false,"explanation":"This describes the co-domain rather than the domain. The domain is the set from which valid input values for the function are chosen."},{"text":"The rules by which output values are calculated.","correct":false,"explanation":"A function’s instructions define how outputs are calculated, not its domain. The domain is the set from which the function’s input values are chosen."},{"text":"The set of all values produced by a function.","correct":false,"explanation":"The values actually produced form the range or image, not the domain. The domain is the set from which valid input values are chosen."}]},{"id":"21a78f18-b7ce-4b6b-9654-c24910aee7d3","cardType":"cloze","points":null,"front":"A [ ] is a category of values (like integer or string); the domain and co-domain are always subsets of these.","back":"A data type is a category of values (like integer or string); the domain and co-domain are always subsets of these.","mcqOptions":[{"text":"data type","correct":true,"explanation":"A data type groups values with shared representation or permitted operations, such as integers or strings. A function’s domain and co-domain may each select subsets of such types."},{"text":"data structure","correct":false,"explanation":"A data structure organises and stores collections of data, rather than naming a category such as integer or string. The required category of values is a data type."},{"text":"parameter type","correct":false,"explanation":"A parameter is a named input in a function definition, not a general category of values. Integer and string are examples of a data type."},{"text":"return value","correct":false,"explanation":"A return value is the result of a particular function application, not a category containing values. The category exemplified by integer or string is a data type."}]},{"id":"2510bcbf-8f4d-4906-8a24-cadc3ee615f5","cardType":"basic","points":1,"front":"What is the result of applying $\\text{add}\\ 4$ where $\\text{add}: \\text{int} \\rightarrow \\text{int} \\rightarrow \\text{int}$?","back":"A new function that adds 4 to its input.","mcqOptions":[{"text":"A new function that adds 4 to its input.","correct":true,"explanation":"The curried type shows that supplying the first integer leaves a function of type $\\text{int} \\rightarrow \\text{int}$. That new function adds the supplied value 4 to its input."},{"text":"A new integer that contains the value 4.","correct":false,"explanation":"Partial application does not add 4 immediately because the second integer argument has not been supplied. It returns a new function that adds 4 to a later input."},{"text":"A new function that multiplies its input by 4.","correct":false,"explanation":"The first application supplies 4 as the first operand, not as a multiplier. Because one integer argument remains, the result is a new function that adds 4 to its input."},{"text":"A type error because only one input is supplied.","correct":false,"explanation":"The function is curried, so supplying only 4 is valid partial application rather than an error. The result is a new function awaiting another integer and adding 4 to it."}]}]}],"examCountdown":{"examDate":"2027-05-11","paperLabel":"Paper 1"},"heading":"Functional programming paradigm","title":"Functional programming paradigm Flashcards","topbar":{"backHref":"/as-level/computer-science/aqa/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"33109ec7-1248-5cb6-b837-1f1744531ca0","walled":"$L39"}],"$L3a","$L3b"]