Exercise 1. For any element x and set A, prove that the sets A and {x} × A have the same cardinality.

Q: 1. Given a relation R on a set A, prove that if R is transitive, then so is R-

A: Transitive means aRb and bRc then aRc

Q: Prove that for any sets A, B, C: (AUB)-B A - B

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Q: Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: Suppose A be any set and R is relation defined on A. Then R is transitive if whenever (a,b)∈R and…

Q: 2. Classify each set as closed, open, or neither. Justify your answer. A= {-+:nEN} E. 1-:] [0, 1)

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Q: 23. With S as in Exercise 21, decide whether or not the set a b U = a is an ideal of S, and justify…

A: According to our guideline we can solve only first question.when you say nothing about any…

Q: How do I write a set T such that S is an element of T and a proper subset of T?

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Q: Let X, Y be two sets. Prove that X × Y C P(P(X U Y )).

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Q: prove for every two sets A and B that A=(A-B)U(A n B)

A: To prove: We want to prove  for every two sets A and B that A=A-B∪A∩B

Q: The subject is on Elements of Set Theory:I really need help and explanation.

A: The symmetric difference of two sets are defined here as,

Q: Prove that if A is transitive set, then A+ is also a transitive set.

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Q: Let A and B be two subsets of E. Prove using the set identities: А - В — В — А.

A: You have asked multiple unrelated questions in the same post. I have addressed the first one. Please…

Q: Suppose A and B are finite sets. Prove or disprove: (aun-aun)- (AUB) = (A u B) ) + Dsjnt(A, B)

A: Consider the equation

Q: Use an element argument to prove that for all sets A and B, (An B) U (AN BC) = A. %3D

A: I have proved that sets on both side are subsets of each other and ten concluded that they are equal

Q: . Let A and B be two connected subset

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Đ A is a closed set.

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

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Q: (iv) Prove that for any sets A, B,C we have A U (B nc) = (AU B)n (AUC)

A: To prove:- Prove that for any sets A, B, C we have AU (B∩C) = (AUB)  ∩(AUC) Method :- I use venn…

Q: Use the element method for proving that a set equals the empty set to prove that for all sets A, B…

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Q: B. Let B and C be the subsets of some universal set X, then prove that (C – B)' = BU C'

A: let B and C be the subsets of some universet x1 then prove that (c-B)' = BUC'

Q: 2. Prove that if A, B,C are sets such that ANBNC = 0, then (A\B)n(C\B) = AnC. Prove that for onu got

A: Recall that, set A\B means a set of elements of A that does not belong to B.

Q: Prove that for all sets A and B, A – B = AN B

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Q: 4. Let A and B be sets. (a) If A and B are subsets of the same universal set U (so A × B is a subset…

A: The definition of complement is given at the end of proof

Q: For the following sets, determine its cardinality.   (ℝ^2)\(ℚ×ℤ)（ is Cartesian product）

A: We find cardinality.     (ℝ^2)\(ℚ×ℤ)（ is Cartesian product）

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

A: We have to show that the following statement is true-  Any open interval (a,b) of the real numbers…

Q: Let A, B, and C be sets. (a) Find |(A U C) \ B|provided |A| = 50, |B| = 45, |C| = 40,|A N B| = 20,…

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Q: 3. Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: We are given  that R is a transitive relation on a set A  Transitive Relation  A relation on a set X…

Q: Prove the following theorem directly. 'Let B, E,W be sets. If B Ɔ W, then BU E WUE"

A: Given B, E and W are three sets .

Q: (b) Let A = (x ER 0<x< 1) and B=(x ER | 0<x<2). Determine whether A and B have the same cardinality.

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

Q: ( Prove that the set {a E R2: a2 ... Prove that the set {a E R2: a2 not equal to 6} is open.

A: Let the set is A=a1,a2∈ℝ2:a2≠6 The objective is to prove that the set A is open.

Q: Let P(S) be the set of all subsets of set S, and let T be the set of all functions from S to {0,1}.…

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Q: 2. Let X be any uncountable set, Prove that: 7. = (GC X: G is countable} U {0} is a topology on X.

A: Since you have posted multiple question... according to company rule we are supposed to answer first…

Q: Let A={(m,n)∣m,n∈N and gcd(m,n)=1} and define the relation ⪯ on A according to (a,b)⪯(c,d)⇔ad≤bc…

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Q: (iv) Prove that for any sets A, B,C we have A U (B nc) = (AU B) n (AUC)

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Q: If X = {u,v} what is the power set of X? (in answering this question, use {} as the null set, and do…

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Q: let X={a,b,c} Construct two topologies for the set that have at least 4 open and that are…

A: Given that X=a,b,c. To construct two topologies that have at least four open sets. We know that in…

Q: Let A and B be two finite sets such that |A| = 20, |B| = 28 and |A U BI

A: Given A=20B=28A∪B=36

Q: (b) Let A = (x ER 0<x< 1} and B = {x ER|0<x<2}. Determine whether A and B have the same cardinality.

A: Solution:-

Q: Let Cc R be a connected set and a , be R. Prove that [a , b] c C.

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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: Prove that a set A is connected iff A is and interval.

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

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Q: Let R and S be reflexive relations over set A. Prove or disprove that RnS is reflexive

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Q: Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)}…

A: Given that A is non empty set which is bounded below by 3. Given that B=x∈R|∃a∈A,b=1a The objective…

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: 6. Let S be a set with a finite number of elements, and let f: S→→ S be a map (a) If fis onto, can f…

A: We are authorized to answer one question at a time since you have not mentioned which question you…

Q: Prove that for all sets A & B, (B-A) = A intersects B^c

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