Show that the following two statements are equivalent: i) the function f : A → B is surjective ii) for every set C and for any functions g : B → C and h : B → C such that g ◦ f = h ◦ f we have g = h.

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

A: We have to prove the given statement.

Q: Let A be a set and let ƒ : A → B be a surjective function. Prove that there exists a subset CCA such…

A: Given: A be a set and f:A→B is a surjective function. To prove: There exists a subset C⊂A such that…

Q: Show that, for any set A, there is exactly one map f from the empty set ∅ to A. When is f injective?…

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Q: Let X,Y,Z be sets and f:X→Y, g:Y→Z. Prove that if g is injective and g◦f is surjective,then f is…

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Q: It follows by definitic ecause F is onto, there exists an x in X, such that F(x) = y image of a set…

A: Everything You have done was correct except the second last step . So, I m doing from there . Since…

Q: For the following relation ~ on set X, determine whether it is symmetric, transitive or reflexive.…

A: For all x in X, x2 ≠ x2 + 1,hence x ~ x Hence ~ is reflexive.  Now consider the equation          x2…

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: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x € R: 1/x e A}. 1…

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Q: Prove that If f,g E R[a,b] with f(x) < g(x) for all xE [a, b), prove that f(x) < Sa8(x).

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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: (d) Using Part (c) and continuity of g, prove that g(x) = mx + l for some m, l e R. (e) Using…

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Q: If for each element y in Set B there exists an element x in Set A such that f(x) = y, then the…

A: A function f : A → B is said to be (a) one to one function - If the images of distinct elements of…

Q: Prove that if f : [-1,1] → R is measurable, then |f| is also measurable.

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Q: a. If f(x,y) is convex with respect to (x,y) and C is a convex set, show that g(x) = infyec f(x,y)…

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Q: Given that f is a function from A onto B, that g :B →C, and that the function g • f is one-one,…

A: In the given question we have to find f and g are one - one function.

Q: Prove whether the following theorem is true or not : If A is any countably infinite set, B is any…

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Q: If X and Y are finite sets, prove that XY = |X|Y. Here, XY is the set of all functions from Y to X.

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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)) = y…

A: If there exists an one-one map from D to C then "|D| &lt; |C|" or "|C| &gt;|D|".

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: Let B be a set and let A ⊆ B be a subset. Suppose that there exists an injective function f : B → A.…

A: Schroder-Bernstein Theorem- Let A and B be arbitrary sets. Using the concept of cardinality. It is…

Q: Let (X, CA) be a measurable space, f: X (-00, ws) and 1: X (-∞, ∞0) are measurable. Prove that the…

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Q: A relation R defined on the given set A will satisfy

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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: Show that if f and g are continuous on [a, b] and if f(a) g (b), then there exists some ce ]a, b[…

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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 f : A → B and g : B C be surjective functions. Prove or disprove whether gof is surjective.

A: Here given that                       f :A→B   and    g :B→C  be both surjective function .  Now we…

Q: 7. If a functionf defined in [a, b] is such that o (i) f is derivable in [a, b], (ii) f' (a) and f'…

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Q: A relation R, is defined on Z, the set of integers by aRb + a - b is even. Show that R is…

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Q: (ii) Let A be a non-empty set and f: AA be a function. Prove that if fof is injective, then / is

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Q: 11. Define a relation P from R to R as follows: For all real numbers x and y with x>0, (x, y)e P…

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Q: Let f(x, y) = domain of f is the set D = {(x, y)|x² + y² > 1}. Select one: 1 1 - x² - y² True O…

A: Given.

Q: Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…

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Q: If a function f is differentiable on [7,11], then which of the following statements must be true?

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Q: Let A and B be sets. Show that               f : A × B → B × A such that                  f(a, b) =…

A: Given:

Q: Suppose A is an uncountable set and B is a denumerable set. Suppose f:A→B is a function. Prove that…

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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: On one side are the two parts of the definition of a function f: X -> Y, the definition of…

A: Definition : A function is an equation for which any x that can be plugged into the equation will…

Q: Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…

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Q: a) Let f : X → Y be a function such that f-1 : Y → X is a closed function, where a closed function…

A: A function is continuous if preimage of open set (in codomain) is open in domain or pre image of a…

Q: Let f : (X,Tx)→ (Y,TY) be a function,where TX is a topology on X and TY is a topology on Y. a) Show…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let f : A B be a function. Prove that f is surjective if and only if, for every pair of functions g…

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Q: Let S be some non-empty collection of real numbers. We know that Vx E S, 3y E S such that x2 y.…

A: In this type of question you need to assume the set of x,y and S that satisfy the condition given in…

Q: 1. For this question, use N as the universal set. Define f = {(X,Y) E P(N) × P(N) : XUY = N and XnY…

A: Given that N(set of Natural numbers) is a Universal set. Also, given is a relation from P(N) to…

Q: Let g be defined on an interval A, and let c ∈ A. (a) Explain why g'(c) in Definition…

A: Given that function g is gefined on interval A and c∈A