编程代考 # Sets module – cscodehelp代写

# Sets module

## `elem x a`

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– Type: `’a -> ‘a list -> bool`
– Description: Returns true iff `x` is an element of the set `a`.
– Examples:
elem 2 [] = false
elem 3 (insert 5 (insert 3 (insert 2 []))) = true
elem 4 (insert 3 (insert 2 (insert 5 []))) = false

## `insert x a`

– Type: `’a -> ‘a list -> ‘a list`
– Description: Inserts `x` into the set `a`.
– Examples:
insert 2 [] = [2]
insert 3 (insert 2 []) = [3; 2]
insert 3 (insert 3 (insert 2 [])) = [3; 2]

## `insert_all xs a`

– Type: `’a list -> ‘a list -> ‘a list`
– Description: Inserts each element from `xs` into the set `a`.
– Examples:
insert_all [2; 3; 3] [] = [2; 3]
insert_all [1; 2; 3] [4; 5; 6] = [1; 2; 3; 4; 5; 6]

## `subset a b`

– Type: `’a list -> ‘a list -> bool`
– Description: Return true iff `a` **is a** subset of `b`. Formally, A ⊆ B ⇔ ∀x(xϵA ⇒ xϵB).
– Examples:
subset (insert 2 (insert 4 [])) [] = false
subset (insert 5 (insert 3 [])) (insert 3 (insert 5 (insert 2 []))) = true
subset (insert 5 (insert 3 (insert 2 []))) (insert 5 (insert 3 [])) = false

## `eq a b`

– Type: `’a list -> ‘a list -> bool`
– Description: Returns true iff `a` and `b` are equal as sets. Formally, A = B ⇔ ∀x(xϵA ⇔ xϵB). (Hint: The subset relation is anti-symmetric.)
– Examples:
eq [] (insert 2 []) = false
eq (insert 2 (insert 3 [])) (insert 3 []) = false
eq (insert 3 (insert 2 [])) (insert 2 (insert 3 [])) = true

## `remove x a`

– Type: `’a -> ‘a list -> ‘a list`
– Description: Removes `x` from the set `a`.
– Examples:
elem 3 (remove 3 (insert 2 (insert 3 []))) = false
eq (remove 3 (insert 5 (insert 3 []))) (insert 5 []) = true

## `diff a b`

– Type: `’a list -> ‘a list -> ‘a list`
– Description: Subtracts the set `b` from the set `a`.
– Examples:
diff [1; 2; 3] [1; 2; 3] = []
diff [1; 2; 3] [1; 4; 5] = [2; 3]
diff [1; 2; 3] [4; 5; 6] = [1; 2; 3]

## `union a b`

– Type: `’a list -> ‘a list -> ‘a list`
– Description: Returns the union of the sets `a` and `b`. Formally, A ∪ B = {x | xϵA ∨ xϵB}.
– Examples:
eq (union [] (insert 2 (insert 3 []))) (insert 3 (insert 2 [])) = true
eq (union (insert 5 (insert 2 [])) (insert 2 (insert 3 []))) (insert 3 (insert 2 (insert 5 []))) = true
eq (union (insert 2 (insert 7 [])) (insert 5 [])) (insert 5 (insert 7 (insert 2 []))) = true

## `intersection a b`

– Type: `’a list -> ‘a list -> ‘a list`
– Description: Returns the intersection of sets `a` and `b`. Formally, A ∩ B = {x | xϵA ∧ xϵB}.
– Examples:
eq (intersection (insert 3 (insert 5 (insert 2 []))) []) [] = true
eq (intersection (insert 5 (insert 7 (insert 3 (insert 2 [])))) (insert 6 (insert 4 []))) [] = true
eq (intersection (insert 5 (insert 2 [])) (insert 4 (insert 3 (insert 5 [])))) (insert 5 []) = true

## `cat x a`

– Type: `’a -> ‘b list -> (‘a * ‘b) list`
– Description: Turns each element of `a` into a 2-tuple where the first element is `x`.
– Examples:
cat 1 [2; 3; 4] = [(1,2); (1,3); (1,4)]
cat 3 [] = []
cat “hi” [1; 2] = [(“hi”, 1); (“hi”, 2)]

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