Let A = {1, 2, 3, 4}. Let F be the set of all functions from A to A. Let R be the relation on F defined by: For all f, gЄ F, fRg go f(1) = 2. (a) Is R reflexive, symmetric, antisymmetric, transitive? Prove your answers. (b) Is it true that for all functions ƒ Є F, there exists a function g € Ƒ so that fRg? Prove your answer. (c) Is it true that for all functions g = F, there exists a function f = F so that fRg? Prove your answer. (d) How many functions f = F are there so that ƒRƒ? Please simplify your answer to a number and provide your recipe.
In order to prove that a statement is false, you must write out its negation
and then prove that the negation is true. Answer counting questions with a detailed recipe
and simplify your answer to a number
Let A = {1, 2, 3, 4}. Let F be the set of all functions from A to A. Let R be the
relation on F defined by:
For all f, g ∈ F, fRg ⇐⇒ g ◦ f(1) = 2.
(a) Is R reflexive, symmetric, antisymmetric, transitive? Prove your answers.
(b) Is it true that for all functions f ∈ F, there exists a function g ∈ F so that fRg?
Prove your answer.
(c) Is it true that for all functions g ∈ F, there exists a function f ∈ F so that fRg?
Prove your answer.
(d) How many functions f ∈ F are there so that fRf? Please simplify your answer
to a number and provide your recipe.
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