Let x E A. Prove that A is an infinite set if and only if A≈ (A - {x}).

Q: Prove that P(XY) = P(X)n P(Y) for all sets X and Y.

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Q: Prove that if alb and alc then a (bx + cy) for all integers x, y.

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Q: Let a and b be integers, at least one of them not 0. Prove that an integer e can be expressed as a…

A: Let a and b be intgers ateast one of them not 0. Our Aim is to prove that an integer e can be…

Q: (c) Let X, Y and Z be sets. Prove that (X\Y)n (X \ Z) = X \ (YUZ).

A: We prove the given statement by using set theory.

Q: 7 Determine whether the set {1, x, x²} is linearly dependent or in- dependent in R (Justify your…

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Q: 1. Let X be a topological space and be an equivalence relation on the elements of X. Define : X→ X/~…

A: Please check the answer in next step

Q: Let ‘E’ & ‘F’ are any two non-empty sets and there order are as follows;…

A: given let ‘E’ & ‘F’ are any two non-empty sets and there order are as follows;…

Q: Let A, = {..., – 2,- 1,0,1,...,i}. Find

A: Ai={..., -2, -1, 0, 1, ......, i} ∩i=1nAi=?

Q: 3. Which of the following statements are true? There may be more than one correct answer. A. Ø CA B.…

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Q: Let A = {1,2,...,n} and consider the functions f: A → A. a. How many functions satisfy f(1) = 1? b.…

A: This question related to function and bijection.

Q: Count the number X of functions y: (20, 21,..., 23) → (5,6,...,8) so that is not onto. X =

A: Given function φ:(20,21,...,23)→(5,6,...,8)

Q: Calculate ∪∞n=1 An and ∩∞n=1 Anwhere for each n ∈N the set An is defined as:…

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Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

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Q: Let v be the set o all pairs of positive numbers (X,y) With two operations + and C denned as: (x1,…

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Q: Exercise 3.3.20. Let A and B be sets. Suppose that BCA. Prove that AxA-BxB = [(A-B) XA]U[AX (A -…

A: We have B⊆A  first we prove that  AxA-BxB⊆  [Ax(A-B)]∪[(A-B)xA]   Then we prove that  AxA-BxB ⊇…

Q: Prove that for any set ACR and any real number a we have m* (A +a)= m* (A).

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Q: II. Let Q-set of all real numbers, A-index set. Identify the generalized union and intersection of…

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Q: Suppose u, v belong to V. Prove that = 0 if and only if ||u|| ≤ ||u+av|| for all a belong to F

A: We have to prove  if and only if

Q: Let x € Z. Prove that if 2 | (x²-5) then 4 | (x²-5).

A: Given that .The objective, if then prove that .

Q: Show that the interval [0,1] is a closed set in R

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Q: 6. Let f: A B be bijective. Prove that f-¹ of = A. -

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Q: Q.No.2 Let X be a universal set and A, B be the subsets of X.Then prove the following. (a) ((A*…

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Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: Banach space. Show that X E and only if X' is reflexive.

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Q: establish a bijection between the power set of a set S and the set of functions S—>{0, 1}

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Q: Suppose a, b € R and a < b. Prove that |R| = |(a,b)| = |[a, b)| = |(a,b]| = |[a, b]|.

A: We first prove that set of real numbers and open interval (a,b) is eqipotential i.e. they have the…