18. What can you deduce about the sets X and Y if you know, a. there is an injective function f: X Y? Explain. b. there is a surjective functionf:X Y? Explain. c. there is a bijective function f: XY? Explain.

Q: Consider sets A and B with |A| = 7 and |B| = 4. How many functions f: A → B are surjective?

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Q: Let A, B, C be sets. Let f : A → B and g : B → C be functions. a) Prove that if f and g are…

A: We shall solve both parts of this question that you have posted.

Q: . Consider the functions f : A → B and g : B → C. a) If f and g are both injective, show that g o f.…

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Q: (a) Show that the following sentence ¬Vz(3yR(x,y) = R(x, f(x, x)) is satisfiable where f is a…

A: a)

Q: Let f(x) = Vr - 5, g(x) = , h(a) = , Find go((gof)+h(x)) at x=-1? 2-9'

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Q: 6.42. Describe the sets in which the functions below are positive. (a) f(x,y) = 1-x?

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Q: 8. Let the functions f: A → B, g:B → C,h: C → D be defined by the figure below. A B h D 3 a. Which…

A: Consider the functions f:A→Bg:B→Ch:C→D

Q: 2. Answer the questions with Yes or No, and then explain why yes or why not. Justify your answer. a)…

A: The solution of given problem is given as follows :

Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

A: According to the given information,

Q: Consider sets A and B with |A|=8 and |B|=5. How many functions f: A→B are surjective?

A: Given: The number of elements in the set A is 8 and the number of elements in the set B is 5.

Q: Prove the following statements a. Let X and Y be nonempty sets and f: X→Y. If A⊆X and B⊆Y, then…

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Q: Let g : B → N × N where B C Z given by g(m) = (|m| + 2, |m| + 3). Choose all subsets B such that g…

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Q: Consider the functions f and g, defined on R. Use set notation to represent the domain of h(x) :…

A: The given problem is related to function and its domain. We need to understand the concept of…

Q: Sketch the following set and determine if it is a domain, open, closed, and bounded or un- bounded.…

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Q: Let F=x y z+A +x'y z'+x'y'z'+x y'z. Suppose simplified form is F=x'z'+x y +x z, What is A =? x' y z…

A: Let us go back from the reverse. That is from the simplified form…

Q: (5) Let ƒ : A → B and g : B → C, where f and g are functions and A, B, and C are sets. (a) Prove…

A: In 2 nd part construct a function

Q: 1. Use (11.20) to check whether the following functions are concave, convex, strictly con- cave,…

A: TO FIND: Whether the functions are concave, covex, stictly convex or neither?

Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

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Q: Write each of the following sets using a defining property (Ax- iom of Specification) and using a…

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Q: Give a proof of the following statement. Suppose A, B and C are sets, f is a surjective function…

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Q: Show that the following two statements are equivalent: i) the function f : B → C is injective ii)…

A: Given: (i)   the function f : B → C is injective (ii)     for every set A and for any functions g :…

Q: Is the function f(x) = x + 1 from the set of integers to the set of integers onto?

A: For onto from set A to B, Every value of set B should exist atleast once in the function.

Q: Let f be the function from {a, b, c, d} to {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d)…

A: For bijection, the function needs to be one to one and onto.

Q: f(x) = 2x2-x + 6 for all values of x %3D

A: Since you have asked multiple questions in a single request, we would be answering only the first…

Q: Let C and D be finite sets and let f :C → D and g : D →C be two functions such that Vy ED f(g(y)) =…

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Q: 2. Let f(x, y) = (2x – y,x – 2y); (x, y) E R × R, (R is set of real numbers.) a) Show that fis one…

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Q: Consider the function θ : {0,1}×N→Z defined as θ(a,b) =a−2ab+b. Is θ injective? It is surjective?…

A: According to the given information it is required to check whether the given function is injective,…

Q: Determine whether Rlle's Theorem can be applied to fon the closed interval. If Rolle's Theorem can…

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Q: . Let R be the set of real numbers and let D be the set of functions d : R x R → R that satisfy the…

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Q: Suppose that A and B are sets and f : A –→ B is a function. Fix subsets X,Y C A. Prove that Imf(X) –…

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Q: let A be a set with n elements and B a set with m elements. How many different functions are there…

A: Concept introduction: Relation: A relation between two sets is a collection of ordered pairs…

Q: (i) Show that F(S) is an infinite set, and that f is injective but not surjective. (i) Is F(S)…

A: iiFS is an infinite set:First we note that S must be an infinite set as no of primes is infinitehere…

Q: Express the set W = {q|q 2} using interval notation. Use "U" for or .

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Q: Consider sets A and B with |A| = 12 and |B|| 8. How many functions f : A → B are surjective?

A: We will find out the required value.

Q: Given x = {5,10,15,20,25,30} and y function from X to Y. Let f(5) = 400, f(10) = 100, f(15) = 500, f…

A: x=5, 10, 15, 20, 25, 30  &  y=100, 200, 300, 400, 500, 600. f5=400,  f10=100,  f15=500,…

Q: Consider the function f(n) = floor(n/2) from Z to Z. a. Is this function one-to-one? Is this…

A: Now, we have fn=floorn2 Part a A function is said to be a one-one function if no two elements from…

Q: Let f(x, y) : = x/Y Evaluate f(3, 4)

A: We need to evaluate f(3,4) To evaluate we need to plug in 3 for x  and 4 for y

Q: 2. If f : A → B and g : B→ C are injective functions, then gof: A → C is also injective.

A: We need to prove/disprove statenemt two.

Q: Let f(x, y) = e*,4 Evaluate f(4, 3)

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Q: Is the function f(x) = x2 from the set of integers to the set of integers onto?

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Q: Let X, Y be nonempty sets. Let A, B c X and P, Q C Y. Let f:X → Y be a function. Prove or give…

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Q: Prove that for any sets C,D⊆Y and for any function f:X⟶Y, f−1(C∩D)=f−1(C)∩f−1(D).

A: We will prove the given statement.

Q: ) Let f: A-B be a function which is not surjective and g : B C which is bijective. Prove that g of:…

A: See full proof below

Q: Please prove the following statement

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Q: What is an injective function from a set A into a set B?

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Q: Which of the following is the set required for F2[x]? 35 - a) {1, (x + 1), (x + 1)²} b) {0, x, x2}…

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Q: f(x,y) = x2 − y from Z × Z → Z. Is this function injective, bijective, surjective, or none of the…

A: The given functions are to be checked for into and onto

Q: Consider the rule f which maps elements in the set A into the set B and which is illustrated below.…

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Q: Let f be the function from {a, b, c, d} to (1, 2, 3, 4} with f (a) = 4, f (b) = 2, f (c) = 1, and f…

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