4. Prove that the sets (–1,5) and [-1,1) U {2} U (3, ∞) have the same cardinality

Q: Do the sets [0, 1] and [0, 1/2) ∪ (1/2, 1] have the same cardinality? Either give an explicit…

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Q: 1. Show that if f is a bijection from a set X onto a set Y then f is a bijection from Y onto X.

A: Please post the multiple questions separately. Here I answered question (1) only as per our policy.

Q: Show that R has the same cardinality as the interval (–2, 2) by building a bijection between the two…

A: This is a problem of Real Analysis and Topology.

Q: 1.4.10. Prove that the closed interval [0, 1] and the open interval (0,1) have the same cardinality…

A: (Schr¨oder-Bernstein theorem) :Let A and B be sets. Suppose there are injective functions f : A → B…

Q: 14. Assume that R is a well-founded relation, x is a regular infinite cardinal, and card{x | xRy} <x…

A: Given That :  R is a well-founded relation, k is a regular infinite  cardinal, card{ x|xRy}&lt;k…

Q: Prove the understated statements concerning the relation &lt; on the set Z of all integers . If    x…

A: x = y  and     O &lt; z

Q: Consider two sets, A = {1, 2, 3, ..., 100} and B = {3, 5, 7, 9, ... , 201}. Prove that |A| = |B| by…

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Q: Suppose that |A| = m and |B| = n. Find the following cardinalities. a. b. Ip(A x B)| c. Ip(A) x…

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Q: Let X be finite set with cardinality=n. Prove that the power set P(X) has cardinality 2^n

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Q: 3.2 Prove that in any metric space (S, d) every closed ball S,ro] is a closed set.

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Q: the sets  [0,1] and  [0,1/2)∪(1/2,1] have the same cardinality, give an explicit bijection to show…

A: We will define a bijection from the set J  = [ 0 , 12 ) ∪(12 , 1]   to the set I =0 ,1   First we…

Q: Prove that [2,5] and (3,4) have the same cardinality.

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Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Two discrete spaces are homeomorphic if there exists a bijection between them and it is possible if…

Q: Consider two fuzzy relations [! 0.1 0.3] R=0.8 0.6 0.9 and S=0.2 0.4 0.9 0 03 0.2 0.5 0.7 0,2 0.7…

A: Let R1 and R2 be two fuzzy relations. The max-min composition of R1 and R2 is defined as follows.…

Q: Consider the non empty sets A and B such that A and B have the same cardinality. Prove, for any set…

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Q: Prove that P(N) is equipotent with the set of functions 2N = {f : N → {0, 1} : f is a function}.…

A: Given that, f:N→{0,1}:f is a function  The aim is to show that, P(N)=f:N→{0,1}:f is a function The…

Q: Prove that: m*(A) 2 0, for all sets A

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Q: 1. Let F be a nonempty family of transitive relations. Prove that NF is a transitive relation. Hint.…

A: Given that the F be a nonempty family of transitive relations.

Q: Let X and Y be two discrete spaces, then! X is never homeomorphic to Y X is homeomorphic to Y if and…

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Q: 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: The relation - on the set A = Z x (N - {0}) by (u, v) - (z, w) if and only if wu = zv. Specify the…

A: We will use the relation wu=zv between two elements (u,v) and (z,w) .

Q: Suppose S and T are infinite sets and d= SoT is a function that is not onto May we conclude that s…

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Q: 5. Construct a smallest binary relation S defined on the set {w,r, y, z} such that S satisfies all…

A: Consider the set, A=w,x,y,z, where all the 4 elements are distinct.

Q: 3.6 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: Suppose S and T are infinite sets and φ: S→T is a function that is not onto. May we conclude that S…

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Q: Show that any open interval (a, b) of the real numbers has the same cardinality as (0, 1). (note…

A: We can use bijection rule to solve the following property

Q: Show that the two given sets have equal cardinality by describing a bijection from one to the other.…

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Q: Let X and Y be two discrete spaces, then * X is never homeomorphic to Y X is homomorphic to Y if and…

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Q: 3. ** Let X, Y be finite sets, and f is a function XY. What can you say about the cardinality if X,…

A: 3. Given X, Y are finite sets and f:X→Y is a function   To Comment about cardinality of X and Y if…

Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

Q: 2.3) Suppose A, C A2 C …· are o -algebras consisting of subsets of a set X. Is U, A¡ necessarily a o…

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Q: df. Every nonempty proper subset of R is either open or closed.

A: False

Q: 5. Suppose that A and B are bounded subsets of R. Prove that AUB is bounded and that sup(AUB) =…

A: Given that A and B are bounded subsets of R.   We have to Prove that AUB is bounded and that…

Q: (a) What is the cardinality of the interval (0, 1), finite, countably infinite, or uncountable? (b)…

A: Schroder-Bernstein Theorem: If there exists injective functions f : A→B and g : B→A between the sets…

Q: 9.C. Prove directly that if K is compact in Rº and F is a closed subset of K, then F is compact in…

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Q: 1. Suppose R and S are symmetric relations on a set A. Prove that Ros is symmetric if Ros=SOR

A: Given:  To prove R and S are symmetric relations.

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

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

Q: Show that the relation R in the set {1,2,3} given by R = {(1, 1), (2, 2),(3, 3), (1, 2), (2, 3)} is…

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Q: Do the sets [0,1] and [0,1/2)∪(1/2,1] have the same cardinality? Either give an explicit bijection…

A: Yes, the sets 0,1 and 0,12∪12,1 have the same cardinality.

Q: e. If x is a subset of V of cardinality 2, then f (x) is the edge connecting the two vertices in x.

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Q: Establish the following cardinality equalities, exhibiting a bijection in each case. Justify your…

A: This question is related to function mapping.

Q: Find the projection of

A: Consider the given vectors, v=111 , u=1-22 Write the vectors v and u in row form v=1,1,1 , u=1,-2,2…

Q: 5. Prove or disprove: for any set A, there exists a relation R on A such that R is both symmetric…

A: Symmetric : If aRb , then bRa , for every a , b∈A Antisymmetric : If aRb and bRa , then a=b

Q: Exercise 4.21. Prove. (1) The set of all functions f : R → R has the same cardinality as : 2R.

A: We'll answer the first question since the exact one wasn't specified. Please submit a new question…

Q: Consider the set X={a,b,c,d} construct all the possible topologies of X that have only three open…

A: Given: The set X=a,b,c,d To find:  all the possible topologies of X that has only three open…

Q: 2. Prove that [0,1] and U-1 An , where A,= [2n, 2n+1] have the same cardinality. %3D

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Q: 4.2 Let Fj, be a closed set for k = 1,2, ... ,n in (S, d). Show that U Fk is closed.

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Q: Prove the understated statements concerning the relation &lt; on the set Z of all integers .   If O…

A: Prove the understated statements concerning the relation &lt; on the set Z of all integers .   If O…