Let A and B be sets where: • A= {x € Z: |x|54}and • Bis a partition of A. Complete the partial definition of B below so that |B| = 4.

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 A and B be subsets of a space X such that A is an open set and B is a closed set. Then a. A-B is…

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Q: Given: f = {(a,b),(b,a),(c,b)} a function from X = (a,b,c) to X a). Write f ∘ f as a set of…

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Q: 5. Let A = {x E Z|x = 12r - 5 for some integer r} and B = {y € Z|y=6s +1 for some integer s}. Prove…

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Q: For A = {1,2,3,4} and B = {3,4,5}, determine Ax B= BxA= X = (A x B) - (B x A) =

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Q: F, there is a disconnection A, B of F such that x E A and y E B. 8.X. If C1, C2 are connected…

A: We use the fact that a set X is connected if there is no continuous function form f from X to Y=…

Q: Prove that Z3 x Zz is isomorphic to Zg.

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Q: Let A={1, 2, 3, 4} and B={2, 4, 6, 8}. Define a function f:A→B such that f(x) = 2x for all x∈A.…

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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: If A is bounded below and B is bounded above, show that inf(A − B) = inf A − sup B

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Q: Prove that the set {x e R : 10/x - x > 0} is bounded. Prove that the set {x E R : x² – 25x > 0} is…

A: To prove that the set x∈R:10x−x>0 is bounded. To prove that the set x∈R:x2−25x>0 is unbounded.

Q: 11. Define F:Z × Z→Z × Z as follows: For every ordered pair (a, b) of integers, F(a, b) = (2a+1, 3Ь…

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Q: Consider the following sets. A = {0, 1}5 A₁ = {20 : x € {0, 1}4} A₂ = {201 : x ≤ {0,1}³} A3 = {2011…

A: A1 = {x0: x∈ {0,1}4}A2 = {x01 : x ∈ {0,1}3}A3 = {x011 : x ∈ {0,1}2}For the above sets to form a…

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: 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 Y be an infinite subset of a compact set XC R. Prove Y' 0.

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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: The Power Set a) Let A = {a, b, c}. Find P(A). b) Let B = {0, 1} Let f: P(A) → B. The input of f is…

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Q: Consider the relation S on N defined by uSy⟺5|(y+u). Is it reflexive?

A: No it is not a reflexive relation

Q: Let A = {2, 3, 4, . . .} be ordered by “x divides y”. (i.) Determine the minimal elements of A.…

A: Let  be ordered by “ divides ”.(i). Determine the minimal elements of .(ii). Determine the maximal…

Q: Let π D₁ = {z € C\{0} : < Arg z < 4 D₂ = {z € C: 0 ≤ Re(2) ≤ ½ & 0 ≤ Im(2) ≤ 1}. a) Sketch D₁ and D₂…

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Q: Let A = {n€ Z| |n|<3} and the set B = {m/k[m€{2,6}, kE {1,2}}. Write the elements of set ANB in…

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Q: Pick a subset from R and call it X. Let your mark on this final examination be x, prior to any bonus…

A: Question: Pick a subset from R and call it X. Let your mark on this final examination be x, prior to…

Q: Exercise 4.4.5. Let A and B be sets. Prove that there is a bijective function f: A × B → BXA.

A: A and B are two sets.  We need to prove there is a bijective function            f : A × B → B × A.

Q: Consider sets A and B with |A| = 12 and |B|| 8. How many functions f : A → B are surjective?

A: We will find out the required value.

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.