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

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: given two sets X-{-1,0,1} and Y={u,v,x,y,z}. There are mappings from X to Y.…

A: The set X=-1,0,1 and Y=u,v,x,y,z

Q: Let F = {Fα : α ∈ A} be a family of compact subsets of R. Prove that an intersection, α∈A, Fα is…

A: A set F is said to be a compact set if it is closed and bounded both.

Q: given two sets X={-1,0,1} and Y={u,v,x,y,z}. There are 125 mappings from X to true or false

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Q: d. Ris a relation on Z* such that (x, y) e R if and only if there is a positive integer n such that…

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Q: Prove that the function g:Z9 → Z, · defined by g(x) = 2x is an isomorphism.

A:

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: Let A = Z+ x Z+. Define a relation R on A as follows: For all (x, y) and (z, w) in A, (x, y)R(z, w)…

A: We will find out the required values.

Q: 1. Show that which of these relations on the set of all functions on Z→Z are equivalence relations?…

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

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Q: A map f:x-y is said to be an open map if for every open set U of X, the set f(u) is open in Y. Show…

A: Given, a map f: X→Y is said to be an open map if for every open set U of X, the set f(U) is open in…

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 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…

A: According to the given information,

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: Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

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Q: let R be the relation on the set A = {1, 2, 3, 4, 5, 6} defined by a R bif and only if a = 2b.

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Q: Let D  be an integral domain and define a relation ~ on DxD such that b is not equal to 0  and d is…

A:

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…

A:

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

A:

Q: Let P be a partition of the nonempty set A. For x, y E A, define xQy if and only if there exists CEP…

A:

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

A:

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 R be the relation on Z defined by    R = {(a,b): a, b ∈ Z, a – b is an integer}.Find the domain…

A: R = {(a,b): a, b ∈ Z, a – b is an integer}. We know that the difference of two integers is always an…

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 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: given two sets X={-1,0,1} and Y={u,v,x,y,z}. There are 3x(3-1)x(3-2)%36 onto mappings from X to Y.…

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Q: Prove that for any set A, the identity function iA is a bijection.

A:

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: Determine if the map g : Q ! Q dened by g(x) = x/(x2+1) is one-to-one.

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: A relation R is defined on R by a R b if a – b e Z. Prove that R is an equivalence relation and…

A:

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

A: Given:

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

A:

Q: Recall that a map has an inverse if and only if the map is 1-1 and onto. Prove that an affine map A…

A:

Q: RUR and RNR symmetric relations on X. are Show that R=Z xZ is an equivalence relation.

A:

Q: Show that linear mapping  T : P2 (R) → R3defined by  T (ax2 + bx + c) = (a + b, a + c, b+ c), is an…

A:

Q: Show that the relation R on Z defined by a R b if and only if 5a − 3b is even, for a, b ∈ Z, is an…

A: Given information: The relation R on Z defined by a relation b if and only if 5a − 3b is even, for…

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: 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 R be the relation on Z→ Z such that ((a,b),(c,d)) eR = a+d =b+c. Show that R is an equivalence…

A: An equivalency relation on a set is a reflexive, symmetric, and transitive relationship for…

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 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: 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