Let A = {1, 2, 3, 4} and B = {a, b, c}. Give an example of a function f: A -> B that is neither injective nor surjective.
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Let A = {1, 2, 3, 4} and B = {a, b, c}. Give an example of a function f: A -> B that is neither injective nor surjective.
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- Suppose f : A → B is an injective function. Show that there is a function g : B → A such thatg ◦ f = idA. Here, idA : A → A is the identity function on A, i.e., the function that satisfies idA(a) = afor all a ∈ A. (Assume f is non-trivial, i.e., A ̸= ∅!)Which functions are one-to-one? Which functions are onto? Describe the inversefunction for any bijective function.(a) f : Z → N where f is defined by f (x) = x4 + 1(b) f : N → N where f is defined by f (x) = { x/2 if x is even, x + 1 if x is odd}(c) f : N → N where f is defined by f (x) = { x + 1 if x is even, x − 1 if x is odd}Given a functor parameterised type m and comes with a function return :: a -> m a, show how functions with the following types can each be defined in term of the other: (>>=) :: m a -> (a -> m b) -> m b combine :: m (m a) -> m a
- A triple (x, y, z) of positive integers is pythagorean if x2 + y2 = z2. Using the functions studied in class, define a function pyth which returns the list of all pythagorean triples whose components are at most a given limit. For example, function call pyth(10) should return [(3, 4, 5), (4, 3, 5), (6, 8, 10), (8, 6, 10)]. [Hint: One way to do this is to construct a list of all triples (use unfold to create a list of integers, and then a for-comprehension to create a list of all triples), and then select the pythagorean ones. def unfold[A, S](z: S)(f: S => Option[(A, S)]): LazyList[A] = f(z) match { case Some((h, s)) => h #:: unfold(s)(f) case None => LazyList() } code in scalaExpand on the concept of Black Box and then go into detail on primivitive functions:For the first problem, in parts b and c, how are the variables being taken from functions that they are not defined in? To clarify, in part c, how is the variable "b" taken from fun2 when it is not defined in that in function. Shouldn't the answer there just be "b(fun1)"
- A function is "almost onto" if it misses exactly one element of its codomain. How many almost onto functions are there from [k] to [n]?Suppose the language, L, over an alphabet Σ0 is regular, and f : Σ0 → Σ1,is a function from the alphabet Σ0 to the alphabet Σ1.The function f can be lifted to words producing the function:ˆf : Σ∗0 → Σ∗1ˆf(ε) = εˆf(aw) = f(a)ˆf(w)Show that the language Lf = {ˆf(w) ∈ Σ∗1| w ∈ Σ∗0} is regular.Hint: You have to use the formal definition of DFAs.Define a function alwaysFollows of type [Integer] -> Bool so that alwaysFollows (x, y, lst) is true if and only if whenever x occurs in the list lst, it is always followed by y. Try your function out on (alwaysFollows('a', 'm', "iamnotaam"), alwaysFollows(6, 9, [1, 4, 6, 9, 8, 5, 6, 9, 7, 3])) ML pls
- Write a function linear_independence that takes a collection of vectors with integer entries (each written as a list), and returns True if this collection of vectors is linearly independent, and False otherwise. Examples: linear_independence([1,2]) should return True. linear_independence([1,3,7],[2,8,3],[7,8,1]) should returnTrue. linear_independence([1,3,7],[2,8,3],[7,8,1],[1,2,3]) should return False.Prove or disprove:(a)∃ sets A,B ∀ functions f: A→B, f is a bijection. (b) if f : A → B is a function, C ⊆ A, and D ⊆ B, then f : C → D is always a function.[Note 1: one is true, the other is false.][Note 2: feel free to draw bubble-arrow pictures here, but make sure you explain what’s going on very carefully.]subject: discrete structures there may be several correct answers Find a minimal expansion as a Boolean sum of Boolean products of the function ???̅+ (?|?̅ ̅̅̅̅̅)? + ???̅̅̅.
