Prove the theorem: For finite sets X and Y, let |X|=m and |Y|=n. If there is an onto function f:X → Y then m > n.

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

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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: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

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Q: Let A, B, C be nonempty sets and let f : AB, g: B → C be functions such that go f is bijective.…

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Q: 2. Let X1 and X2 be sets. Define functions T1 : X1 × X2 → X1, T2 : X1 × X2 → X2 by T| (X1, X2) = x1,…

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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: Prove: Let X and Y be two topological spaces. Then the function f:X →Y is continuous if and only if…

A: This is a problem of topology.

Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: Let D, E be non-empty sets, let ACD, let BCE, and let f: D E be a function. Prove that Anf¹(B) ≤…

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Q: (b) Let F be a non-empty subset of R. Prove that if OF C F, then F is closed.

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Q: Let X = {1, (b) xm} and Y = {y₁, Yn} be finite sets. Determine the number of one to one functions…

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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: Let B be the set of all bounded sequences of real numbers and define the function d: B xB +R by d(x,…

A: In the above question it is given that B is the set of all bounded sequences of real numbers and a…

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: Let f be the function from {a, b, c, d} to {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d)…

A: For bijection, the function needs to be one to one and onto.

Q: Let X and Y be sets with A,B ⊆ X. If f : X →Y is a function, prove that f(A∪B) =f(A)∪f(B)

A: I am going to solve the problem by using some simple algebra to get the required result of the given…

Q: For any set X with at least 2 elements, define a function f : P(X) x P(X) → P(X) by f(A,B) = A △B…

A: We have taken two different ordered pair of subsets of X and shown that their image is same. Hence f…

Q: a. Prove that for any sets A,B⊆X and for any function f:X⟶Y, f(A∩B)⊆f(A)∩f(B).       b. Prove that…

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Q: Let A be a nonempty set and let f: A -> A be a function. Prove that if fof = iA, then f is…

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Q: Show that for every pair of sets X and Y, the sets X × Y and U {(x, y) | y ≤ Y} xEX are equal.

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Q: Let f(x, y) = domain of f is the set D = {(x, y)|x² + y² > 1}. Select one: 1 1 - x² - y² True O…

A: Given.

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

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Q: Prove that if f: R → R is continuous, then the image of a bounded set over f is bounded.

A: We need to prove that, if  is continuous, then the image of a bounded set over  is bounded.We know…

Q: ACX A + Ø and A + X. Let B = AU fr(A). .show that for the topology T = {X,Ø, A, B} it happens that…

A: Given A⊂X where X is non-empty and A≠X Suppose, T=X,∅,A,B be a topology.  Here B=A∪frA To prove for…

Q: Suppose that f is holomorphic in an open set N C C. Prove that f is constant if either one of the…

A: (a)  Let f(z)=u+iv. Then R(f)=u. Given that R(f) is constant .Then u=c ,a constant . Then ux=uy=0.…

Q: Let A be a nonempty set. Prove that if there is an injective function f :(0,1) -> A, then A is…

A: We will prove the above statement by contradiction. by taking the assumption that A is countable…

Q: Let A be a nonempty set and let f : A → B be a function. Prove that f is one-to-one if and only if…

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Q: Prove that for any sets C,D⊆Y and for any function f:X⟶Y, f−1(C∩D)=f−1(C)∩f−1(D).

A: We will prove the given statement.

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

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Q: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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Q: # 2. Prove that for all sets A and B, (A - B) N B = Ø. Proof (A Proof By Contradiction):

A: The given problem is to prove the following statement using the method of contradiction. To prove :…

Q: Prove or disprove. Given any set S with any surjective function f: S → S, f must be bijective.

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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”.