Let A be a non-empty set and f.g:A→A be any functions such that fog is the identity map of A. Then

Q: Let R = - b ,be Z}and let ø : R→Z be a mapping defined by : b = a-b. a a

A: We will find out the required value.

Q: Let A and B be topological spaces and endow A x B and B x A with respective product topologies. Show…

A: Homeomorphism: Let X and Y be topological spaces; let f:X→Y be a bijection. If both f and f-1:Y→X…

Q: Let B {(x,y, z) E R: -1< y +x² < 1, 0 < z – r' < 2} CR. %3D (a) Prove that B is a closed subset of…

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Q: Let A={a,b,c,d,e}. Find a relation R on A which is symmetric and transitive but not reflexive

A: Reflexive relation means every term of a set is related with itself

Q: then L is a linear operatorE Let L(x): R R be a mapping such that L Select one: True False

A:

Q: Let X and U be Banach spaces, let C: X→ U be a compact linear map, and let xnx in X. Show that Can →…

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Q: Let m, n e Z4 and X + 0. a. If m < n, define an injective map f : Xm → x". b. define a bijective map…

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Q: Show that the map T : R3 -→R? given by T (x1, X2, X3) = (x1 + x2, X3) is an open map.

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Q: Let ƒ be a mapping from [2, +∞ [to, +∞[defined by: 3 f(x) = x + = x f has a fixed point. O False O…

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Q: then L is a linear operator. Let L(x). R R be a mapping such that L( X2 Select one: O True O False

A: We will check whether the given statement is true or false.

Q: Letf, g, and h be mappings from A into A such thatf0 g = h 0 g. Thenf = h.

A: given g be mappings from A into A Thus When x∈A⇒gx=k∈A for each x Now Also given  f, g, and h be…

Q: (1) Let T: R3 → M2x2 be a linear map that is injective. Then T is also surjective.

A:

Q: Suppose A is a function that assigns to each subset of the plane a number in such a way that the…

A: Given that AX≥0 and if X and Y are disjoints then AX∪Y=AX+AY To prove if X⊂Y then AX≤AY It is known…

Q: An open map is a map f : X → Y such that for every open U C X, the set f(U) is open in Y. Prove that…

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Q: Define a map t: R × R → R × R by t(a, b) = (a+ b, a - b). Prove that t is a one-to-one…

A: The solution is given as

Q: Given any point a E E, any point be E', and any E, the map f: E→ E' defined such that f(a+v)=b+h(v)…

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Q: Let R be the relation on N defined by x Ry if x and y share a common factor other than 1. Determine…

A: The relation R is defined by xRy if x and y share a common factor. We need to check reflexivity and…

Q: Let k e Z+. Let A be a set such that |A| = k + 1, and let a e A (i.e., a is a fixed but unspecified…

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

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Q: Let 7 and r' be two topologies on a set X and let i : (X,r')(X, T) be the identity map. A) r' is…

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Q: Prove the following: Theorem. Assume that f: A → B and g: B → A are injective maps. Then there…

A: Given:f: A→B     injective mapg:B→A     injective mapTo Proof:There exist bijective map h: A→B.…

Q: Let m, n E Z4 and X # Ø. (a) If m < n, define an injective map f : X™ → x". (b) Define a bijective…

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Q: Let f : [a, b] → R be a two-to-one map, i.e., for every y E Im(f), there are exactly two points a1,…

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Q: Let p: GL(2, R) → R' be a homomorphism map defined by p(A) = det(A), then %3D a) Ker o = GL(2, R) b)…

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Q: The image of the line Im(z) = -2 under the mapping f(z)= i(z) is

A: Solution:

Q: (b) Every continuous map S? →T² is homotopic to the constant map.

A:

Q: Let X be a Banach space and T = L(X, X) have ||T|| < 1. Define Tº to be the identity map (that is,…

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Q: Question 4. Let f:X→Y be a mapping from a topological space (X,7x) a topological space (Y,ry). Prove…

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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 f: X→Y be a surjective function. Define a relation on X by setting ay iff f(x) = f(y). Let E be…

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Q: . Show that every map f : A → B of sets decomposes as hog where g : A → C is surjective and h : C →…

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Q: Let F:G - G/K be a mapping defined as follows: F(a) = a + K for all a E G. Then O Fis monomorphism O…

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Q: Let X be any set and let Y = 27X - {x : X {0, 1}}, that is, Y is the set of all functions from X to…

A:

Q: Let R be the relation on N defined by x R y if x and y share a common factor other than 1. Determine…

A: The relation R is defined as xRy on natural numbers Nsuch that x and y share a common factor other…

Q: Every continuous map T2 –→ S2 is homotopic to the constant map

A: To prove: Every continuous map from the Torus T2 to the sphere S2 is homotopic to the constant map.

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

A: Given:

Q: a Let R = . b DeZ and let o: R→Z be a mapping defined by: b = a-b. b a

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Q: . Let R be the relation on N defined by xR y if x and y share a common factor other than 1.…

A:

Q: Let f: X - Y and g : Y - X be maps between sets. Suppose that g•f is equal to the identity map idy :…

A:

Q: Let L and M be two continuous linear maps such that the composite Mo L is defined. By quoting…

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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: Suppose q: X →Y is a quotient map and that it is one-to-one. Show that q is a homeomorphism. (Hint:…

A: Suppose q: X→Y is a quotient map and that it is one-to-one. Show that q is a homeomorphism.

Q: Let V: U → R be a strict Lyapunov function, where U CR" is a neighborhood of a compact and invariant…

A: Let V:U→R be a strict Lyapunov function, where U⊂ℝn is a neighborhood of a compact and invariant set…

Q: Let B= ((p.4) e R p + =1}. (n) Show that B is NOT disconnected (b) Given that there is no continuous…

A: Given: B=p,q∈ℝ2 : p2+q2=1. There is no continuous mapping f:B→ℝ. To prove: B is not disconnected.…

Q: Suppose q: X → Y is a quotient map and that it is one-to-one. Show that q is a homeomorphism. (Hint:…

A: Given: q:X→Y is a quotient map and it is one-to-one. To show: q is homeomorphism.

Q: Let p : X Y be a quotient map. Let Z be a space and let g : X→Z be a continuous map that is constant…

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Q: Let AC R " be a measurable set and let a € R. Using the transformation theorem , show v ( a + A ) =…

A: This is a problem of measure theory.