Let f: R -> R be continuous and X ⊆ R be connected. Prove f(X) is connected.

Q: Exercise 16. Prove the above theorem. Hint: show A ∪ B = A ∪ B − (A ∩ B). Theorem 35…

A: To prove: A∪B=A+B-A∩B

Q: Let A be an arbitrary set. Show that there does not exist a bijective function f: A → P(A). = Hint:…

A: As per the guideline, I am supposed to solve only first question from the given problems.

Q: topological space X is the intersection of all its closed neighbourhoods. A regular if and only if…

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Q: 1. Give an example of a set S with a relation ~ on S such that ~ is symmetric and transitive but not…

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Q: Let A, B, C be sets. Let f : A → B and g : B → C be functions. a) Prove that if f and g are…

A: We shall solve both parts of this question that you have posted.

Q: Let X be a topological space. Show that the following are equivalent. (i) X is disconnected. (ii) 3…

A: Let X be a topological space then, To prove: Following statements are equivalent: (i) X is…

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: Prove that the set A = {x ∈ R | 3x + 2 ≤ 0} has a maximum

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Q: e) Does the equality |A U B| = |A| + |B| hold for every set A and B? f) Is it true that for any…

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Q: the collection of all close subsets of R is called the topology of R

A: To check: Whether the statement is true or false: Given: The statement is "The collection of all…

Q: Let ∼ ⊆ X ∗X be an equivalence relation on a set X. For x ∈ X define ̄x := {y ∈ X |x ∼ y}. Then it…

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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: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: Is it possible for a relation R on a set X to be both symmetric and antisymmetric? If so, give an…

A: Let R be a relation on set A .then SYMMETRIC relation: ANTISYMMETRIC relation: Recall: Identity…

Q: Let A be a bounded non-empty set. Then inf A < sup A. True False

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Q: Determine whether or not the relation R on the set of all people is reflexive, netric,…

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Q: Let ~ CX *X be an equivalence relation on a set X. For x EX define x := := {x |x E X} is a partition…

A: This is a question from Relation.

Q: Let X be a set and P(X) its power set. Explain what it means that the cardinality of P(X) is…

A: It is required to explain the meaning and to prove that the cardinality of power set P(x), of X, is…

Q: Say a set A is countable if there is an injective map f: A → N. Show that the product of countable…

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Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

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Q: If X is a Hausdorff topological space, is the quotient space X/∼ necessarily Hausdorff? Explain.

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Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

Q: Prove that T is the discrete topology for X iff every point in X is an open set

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Q: Exercise 4. A subset A of a topological space X is called dense if cl A = X. The set Q of rational…

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Q: Let A and B be open sets in a topological space. Is A ∩ B open?

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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: Let B be a set and let A ⊆ B be a subset. Suppose that there exists an injective function f : B → A.…

A: Schroder-Bernstein Theorem- Let A and B be arbitrary sets. Using the concept of cardinality. It is…

Q: 15. Let A be an infinite set. Prove that there exists an injective function f: N→ A.

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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: Let X = [0, 2] ⊆R, R = {(x, x)||x ∈ [0, 1]}∪{(1, y)|y ∈ [1, 2]}⊆[0, 2] ×[0, 2] What properties does…

A: It is given that, X = 0, 2 ⊆ ℝ and  R = x, x : x∈0, 1∪1, y : y∈1, 2 be a relation on 0, 2 We have to…

Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

Q: Suppose E is a bounded, nonvoid set of real numbers. Define |E| to be the set { x: x = E}. Is it…

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Q: 2. Answer the questions with Yes or No, and then explain why yes or why not. Justify your answer. a)…

A: Answer :

Q: (b) Let {A₁, A2,...} be a partition of a set X, and R the relation {(x, y) = X²: there exists j such…

A: An equivalence relation is a binary relation between two elements of a set that satisfies three…

Q: Cardinality: Let A and B be sets. We write A B (and say A has the same cardinality as B) if there is…

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Q: Provide examples of sets A and B where strict inequality sup(A+B) < sup A+ sup B holds.

A: We have to given an example to prove given inequality:

Q: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

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Q: Prove or disprove: All uncountable measurable subsets of ℝ have positive measure.

A: All uncountable measurable subsets of ℝ have positive measure.