2. Let f : X → Y be onto. For each b e Y, let A, {As : b € Y} is a partition of X. f6]. Prove that

Q: Take any r > 0 and any c ∈ R. Show that the r-ball Br( c ) is an open subset of R.

A: We need to prove that for any r > 0 and any c ∈ ℝ  the r-ball Br( c ) is an open subset of ℝ.

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 X:= (a, b.c), n=(1,2,3.4), Ty:= (0,Y, {1,3), [3,4}, {3), [1,3,4}}. Let fx-Y. f:= {(a,3). (b, 3).…

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Q: H is a Hilbert space, A: H → H is linear ar = for all r, y e H. Use th closed graph theorem to show…

A: For two normed spaces V and W, a linear operator T:V→W is closed if for each xn∈V, xn→x and Txn→y…

Q: Let D be a subset of R. Show that f : D → R is uniformly continuous on D if and only if ƒ satisfies…

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Q: Let A be an open subset of R. Is it true that (A)° = A?

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Q: Let f : A → B be a function, {B;}jej a family of subsets of B. Prove that f-'(Njej B;) = Njej…

A: Given that f : A → B is a function. Also given that Bjj∈J is a family of subsets of B . We have to…

Q: Y= {1, 2, 3} and  Z={k, e}. How many onto function are there from Y to Z

A: Formula: If X has m elements and Y has 2 elements, the number of onto functions will be 2m-2 So in…

Q: Show that if f: A→B and E, F are subsets of A then f(E∪F) = f(E)∪f(F) and f(E∩F) ⊆ f(E)∩f(F).

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Q: -> 2. Prove: Suppose that a function f: [a, b] → R is bounded and P = {a = x0, x1, x2, ..., xn = b}…

A: The objective of this question is to prove that for a bounded function f defined on a closed…

Q: 10. Let (X, Tx) and (Y, Ty) be two topological spaces, such that X = {a, b,c, d}, Tx = {0, X, {a},…

A: Definition: A function f is continuous from X to Y if inverse image of each open set in Y is open in…

Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: Let f: R -> R and E ⊆ R. Prove f is continous iff f-1(E) is closed whenever E ⊆ R is closed

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Q: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection ?…

A: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection?…

Q: Let D be a subset of R. Show that f : D → R is uniformly continuous on D if and only if ƒ satisfies…

A: According to the given information, it is required to explain that:

Q: The continuous image of a connected set is connected. That is, if X is connected and f: X→ Y is…

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Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

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Q: Let E = {0, 2, 4, 6, ...}, define ƒ: E × E → Z¹ by the formula f(m, n) = 4m5¹, \ (m, n) € Ɛ × Ɛ.…

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Q: (b) Let F be a non-empty subset of R. Prove that if OF C F, then F is closed.

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Q: Let :Q-21 → be an epimorphism. Hand written plz a) Show that f(n)-n for every n e Z. b) Deduce that,…

A: Here f:ℚ→ℚ is an epimorphism. That is f is a surjective ring homomorphism from ℚ to  ℚ. Now we can…

Q: Let X = {o, p, s, w}, Y = {a,b, e, g, k}, and Z = {1, 2, 3, 5, 7, 9}. Let ƒ : X → Y where f = {(o,…

A: X=o,p,s,wY=a,b,e,g,kZ=1,2,3,5,7,9f:X→Yf=o,a,p,g,s,e,w,bg:Y→Zg=g,3,e,2,b,9,k,7,a,1

Q: Suppose f: A → B, f is one-to-one and onto, and Y c B. Show that the inverse image of Y under f and…

A: The function f is one-one and onto.   The function f is one-one if two different element in A…

Q: Prove that the following relation from Z= {c, o, u, n, t, r, y} to X= {h, a, z, k}is a onto but not…

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Q: Give the relation R induced by the partition above on A.

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Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: 1. Let X be a set and f: X X be a function. Suppose that fofof is an injection. Prove that f must be…

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Q: Let X = {1,2,3,4} and Y = {5,6,7}. State whether a function f: X → Y can be one-to-one or not…

A: By using definition of one-time and onto function we solve the given problem :

Q: Let kEZ". Let A be a Set svch that JA-RtI, Und RT aLA いt u fixed but uns pecified element of A).…

A: Given that fX=X∪a.

Q: (#4) Show that a set SCR" is closed if and only if bdry(S) CS.

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Q: a. Write o using the two row form. b. Write o using cycle notation. c. Write the function o¬1 in the…

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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: 4. Define f:R → R, define f (x) = 3x + 5 for all x e R. Show that a. fis one - to - one; b. f maps R…

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Q: a) Supposef:A→N is a function such that for every k∈N, there are exactly two elements x in A such…

A: Let  f:A→ℕ be a function. For every k∈ℕ, there are exactly two elements x in A such that f(x)=k. For…

Q: 9. Let F : R" → R". Prove that for each a the mapping h→ dF[a; h] is linear on R".

A: Given: F:ℝn→ℝm   To prove: For each a, the mapping h↦dFa;h is linear on ℝn

Q: Let S = {1,2,3, 4}. For each of the following relations on S give its transitive closure. a) {(1,1),…

A: Concept: In mathematics, the transitive closure of a binary relation R on set X is the smallest…

Q: 7. Show that the function f : R → R* defined by f(x) is not onto. x2 +1

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Q: 4. Let X be an ordered set, z E X, and SC X. Prove that x = sup S, if and only if Vy EX (yy).

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Q: Let X be a finite set and f: X → X. Prove that f is an injection iff f is a surjection. (Hint: If…

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