Let X, Y be two sets. Prove that X × YC P(P(X U Y )).

Q: (b) Consider the set R. Prove: (i) VxER, x 0 = 0; (ii) If x y = 0 then either x = 0 or y = 0, or…

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Q: Let A and B be subsets of topological spaces X and Y, respectively. Prove that CI(A × B) Cl(A) ×…

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Q: 3. Let r>0 and define the closed ball by B,(z) = {y € X : d(x,y) Sr} Show that B,(r) is closed set…

A: Given X be a metric space with metric d. Now define a topology τ on X as follows: τ=B⊆X:…

Q: Let (X,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

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Q: Consider the set P = {f€C([0, 1], R) | ƒ is nondecreasing} Determine if (P,d2) is separable. Here,…

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Q: Let X,Y,Z be sets and f:X→Y, g:Y→Z. Prove that if g is injective and g◦f is surjective,then f is…

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Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

A: We have, A=set of positive numbers  A is bounded above set so, for all  x∈A ∃ K∈ℝ such that x≤K We…

Q: 4).ェ Pare subset, -f y such that CeD than fic) is not eD, then the Convers %3D そイue.

A: Solution As it's mention C=D and f^-1 same , so each and every element from  f inverse we get equal.…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 9 an…

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/xe A}. 1…

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Q: Find the Cartesian Product of the given sets:  With A = {c, d, f} and B = { w, x}, find B x A.

A: A = {c, d, f} and B = {w, x}, B x A = { (w,c), (w,d), (w,f), (x,c), (x,d), (x,f) }

Q: Let R have its usual topology and let X = {a,b, c}. Define f : R –→ X by а, х

A: Given, ℝ have its usual topology and let X=a, b, c. Define f: ℝ→X   by…

Q: If X and Y are finite sets, prove that XY = |X|Y. Here, XY is the set of all functions from Y to X.

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Q: 3. Let r>0 and define the closed ball by B,(1) = {y €X : d(r,y) Sr} Show that B,(r) is elosed set…

A: Given that X be a metric space with metric d. The given topology on X is defined as:…

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

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Q: Let C and D be finite sets and let f: C D and g: D C be two functions such that Vy eD f(g(y)) = y…

A: If there exists an one-one map from D to C then "|D| < |C|" or "|C| >|D|".

Q: Let A=(0, π ) υ (π e?] %3D c. Is A a closed set? d. Does A contain any isolated points? е. Find the…

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Q: Consider the sets X = {(x, y) E R²: x² + y² ≤ 1) and Y = ((x, y) = R²: x ≥ 0]. Sketch on R²the sets…

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Q: Consider the relation S on N defined by uSy + 5|(y + u). Determine whether S is symmetric, reflexive…

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Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: Let r = ⟨x, y, z⟩ and r0 = ⟨1, 1, 1⟩. Describe the set of all points (x, y, z) such that r − r0 =…

A: Given: r=x,y,z and r0=1,1,1 To describe the set of all points x,y,z such that r-r0=2 Solution:…

Q: Prove ¬(r ∨ ¬(r ∧ s)) is logically equivalent to F using equivalence laws

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Q: Let A be a non-empty set of positive numbers, which is B={re R:1/xe A}. Prove that B is bounded…

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Q: (f) The set B = {(z, y) E R² : ² + y < 1} ; is disconnected. as a subset of R? with indiscrete…

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Q: Let X, Y be sets and f : X →Y. Prove f"(A\B) = f"(A) \ f¯"(B). for any A, B CY.

A: Solution:Given:          X, Y    are the Sets and                      f: X→ Y To prove:  f-1(A \…

Q: Prove: If U is a neighborhood of x0 and U ⊂ V, then V is a neighborhood of x0. Also, if U1 ..., Un…

A: The neighborhood of a point: Let (X, T) be a topological space and x0∈X. A neighborhood of x0 at is…

Q: Prove that the boundary points of an open interval (a, b) CR are the points a and b.

A: A point a∈ℝ is said to be boundary point of a set A⊂ℝ, if Ba,r∩A≠∅ and Ba,r∩Ac≠∅ for every…

Q: Let V C(R ) be a nonempty open set satisfying a – b EV whenever a, b e V. Prove that V = R. %D

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Q: Let A be a non-empty set of positive numbers, which is unded above and let B = {x e R: 1/xe A}. 1…

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Q: Let X be an uncountable set and let T = {U C X :U = Ø or U¢ is finite} %3D Then is topological space…

A: let X be an uncountable set and T=U⊑X : U=∅ or Uc is finite i.e cofinite topology on X. Then the…

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}. 1…

A: Remark: Completeness property of R: Every bounded above non-empty subset of R has a least upper…

Q: Let X be a finite set and f: X → X. Prove that f is an injection iff f is a surjection. (Hint: If…

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Q: Find the intersection of the sets. {w, y, z}∩∅

A: Given, w,y,z∩∅we have to find intersection of the sets.

Q: Let f : (X,d) - (Y, d1) becontinuous fumction, then for any closed set G in X, f(G) is not necessary…

A: False

Q: Sketch the following sets of points in the x-y plane. {(x, x^2): x is in R}

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Q: Let X = R, and M = {0}, then span (M) = %3D %3D OR O {0}

A: If, M = {0} then, span(M) = {0}

Q: prove with a table if x → y then xor y xor y = y

A: Please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Given the set T := {(x, y) ∈ R^2 : |xy| Less or equ 1}. Is T a compact set? Show your working. If…

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Q: Assume that ∀x∃yP(x, y) is false and the domain of them is nonempty. Which of the following   must…

A: See the detailed solution below.

Q: The relation xy >= 1 where x,y are on the set of all integers is reflexive. True False

A: Given relation xy≥1, where x,y are on the set of all integers

Q: Assume that ∀x∃yP(x, y) is false and the domain of them is nonempty. Which of the following must be…

A: Option (III) is correct.

Q: let E be a set that baunded. Prove that Inf E = Sup -E) where R, XE E y

A: To prove for a buonded set E inf(E) = -sup(-E) , where -E = {-x, x belongs to E} , we first prove…

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

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Q: Let S be some non-empty collection of real numbers. We know that Vx E S, 3y E S such that x2 y.…

A: In this type of question you need to assume the set of x,y and S that satisfy the condition given in…

Q: By algebraic properties of R, prove that for every x, y € R with y # 0 (-x) y

A: We have to prove that by algebraic property that ∀x,y∈ℝ with y≠0 -xy=-xy

Q: Sketch the set of all points (x, y) such that x + y| < e*.

A: Consider the function x+y≤ex. An absolute function is defined as x=x if x≥0-x if x&lt;0. So, there…

Q: Consider the set R. Prove: (i) VxER, x 0 = 0;

A: To prove: For any x∈ℝ, x·0=0.

Q: Let x1 and x2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: Assume f,g : [a, b] → [c, d] where a < b and c < d. Prove that sup{f(x)+g(x) : x € [a, b]} <…

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