Let f: A --> B and g: B --> C be maps. Show that if g of f is 1-1 and onto, then f is 1-1 and g is onto.

Q: Prove that the mapping f: G- G defined by f(x) = x-l is an isomorphism of G onto itself.

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Q: Prove that the mapping from G O H to G given by (g, h) → g is a homomorphism. What is the kernel?…

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Q: Let D be an integral domain. Define a ~ b if a and b are associates.Show that this defines an…

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

A: Given f:X→Y and g:Y→X be the maps between the sets.  Suppose that g∘f=idx. Then We need to prove or…

Q: Let f: X → Y and define x~y if f (x) = f (y). Show that - is an equivalence relation on X.

A: The objective is to show that ~ is an equivalence relation on X.

Q: Let - be the relation on R\{0} given by x ~ y if and only if xy is positive. Prove that this is an…

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Q: - b if and only if f (a) = f(b). Show that ~ is an Let f: A → B be a surjection. Define a ~…

A: Prove ~ is an equivalence relation. 1) Reflexive Since f is a function, f(a) = f(a) and so a ~ a for…

Q: 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

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Q: Show that the relation R in the set A of all the books in a library of a college,given by R = {(x,…

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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: 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: Prove that "is similar to" is an equivalence relation on Mnxn(F ).

A: We have to prove that “is similar to” is an equivalence relation on Mnxn(F ) i) Every matrix A is…

Q: Define a relation on Z by x ~ y if and only if x + 2y is divisible by 3. Show that this relation is…

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Q: Let k: Y → Z to be a map; • Prove that if fo, fi: X → Y are homotopic maps, then ko fo, ko fi: X Z…

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Q: Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following…

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Q: Show that the set A = {fm,b : R → R | m # 0 and fm,b(æ) group under composition of functions. (The…

A: To Show- Show that the set A = fm,b : R → R | m ≠ 0 and fm,bx = mx + b, m, b ∈ R forms a group under…

Q: If E is an extension of F and f (x) e F[x] and if o is an automorphism of E leaving every element of…

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Q: Let R = { ( a,b ) | a,b ∈ Q and a − b ∈ Z } . Prove that R is an equivalence relation on Q

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Q: 4) Let 7 be the Euclidean topology on R and 71 be the discrete topology on R. Define f: (R, 71) →…

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Q: a Define a relation R on N by (a,b) e Rif and only if EN Which of the following properties does R…

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Q: (b) Let - be the relation on C defined by z ~ w if and only if z + w e R. Proye that is an…

A: This can be solved as follows:

Q: If E is an extension of F and f (x) e F [x] and if o is an automorphism of E leaving every element…

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Q: 3. Define the relation R on Z by mR, if and only if 2|(m – n). Show that R is an equivalence…

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Q: Let X = R. Define an equivalence relation ~ on X by x ~ y if %3D and only if x – y is an integer.…

A: Let X, ?X be a topological space, and let ~ be an equivalence relation on X. The quotient set,…

Q: Let ∼ mean “is isomorphic to.” Prove that ∼ is an equivalence relation on the class of vector spaces…

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Q: Let R be a relation on Z. Then Ris an equivalence relation if it is defined by xRy if and only if…

A: R is an equivalent relation iff R is reflexive. R is symmetric. R is transitive.

Q: Define a relation on Z by a R b if a² = b². a. Prove that R is an equivalence relation. b. Describe…

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Q: Suppose f, g, and h are all mappings of a set A into itself. a. Prove that if g is onto andf 0 g = h…

A: Given that, f,g and h are all mappings of a set A into itself. If g is onto and f∘g=h∘g  To prove…

Q: Prove or disprove: for any set A, there exists a relation R on A such that R is both symmetric and…

A: Let  A be any we define a relation R on A s.t

Q: 4. Let R be the relation xRy if 3| (x – y) on Z. Prove that R is an equivalence relation.

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Q: Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following…

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Q: Let X be the set of all subsets of ℤ . Define a relation R on X as follows: Given two subsets of…

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Q: Let C = {a, b, c, d}. Find a relation R on C that has exactly 3 ordered pair members and is both…

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Q: Let A = {-5, -4, −2, 0, 3, 6, 8), and define an equivalence relation R on A as follows: (x, y) E R…

A: Let A = {-5, -4, -2, 0, 3, 6, 8}, and define an equivalence relation R on A as follows: (x, y) in  R…

Q: Show that which of these relations on the set of all functions on Z-→Z are equivalence relations?…

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Q: Give examples of relations R and S on Z such that  Both R and S are not equivalence relations but R…

A: We need to find two set which are on equivalence on Z but their sum do.  LEet, R= a<b S=a>b…

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

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Q: (a) Every continuous map S2 → RP² is homotopic to the constant map.

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Q: Show how a projective transformation in â: R°/ → R°/~ induces an affine map a if and only if ââ maps…

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Q: Define a relation R on Z by declaring that xR y if and only if x^2 ≡ y^2 (mod 4). Prove that R is…

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Q: Define a relation T on R as follows: for all x and y in R, x T y if and only if x2=y2. (a) Prove…

A: Given that a relation T on R such that for all x and y in R, xTy if and only if x2=y2 (a) We have to…

Q: 18. Let R be the relation on R defined by xRy → x – y is an integer. Prove that R is an equivalence…

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Q: Let R be the relation on Z defined by x Ry if and only if x+ 3y is even. Prove that R is an…

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Q: Let R be a relation on Z. Then R is an equivalence relation if it is defined by xRy if and only if O…

A: A relation R between the elements of a set is said to be equivalence relation if it is: Reflexive…

Q: For a,y E R, let z~y if and only if (x - y) € Q. Show that ~ defined as such is an equivalence…

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Q: Let S, T, and R be sets. Assume that f: S-» T, and g: T -» R are maps Assume that gf is onto. Does f…

A: Assume that gf: S→R is onto,Then, by the definition of onto:

Q: Label each of the following statements as either true or false. Letf, g, and h be mappings from A…

A: To label the given statement