) Prove that the set An := {T € Sn | sign(r) = 1} has n! elements for all n > 2. Show this by constructing a suitable bijective mapping between An and Sn \ An . Remark: Without justification, you may use the fact that if there is a bijective mapping between two (finite) sets, then these must have the same number of elements.

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: Consider the relation R on the set Z defined by a Rb if |a| = |b|. Determine if R is reflexive,…

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Q: Define relation R on (Z+ U {0}) × Z+ by (a, b)R(c,d) if ad = bc. Describe the equivalence class [(2,…

A: Our objective is given below:

Q: Assume that you know that ~ is an equivalence relation on R\ {0} defined as follows: xy > 0. Find…

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Q: Let R be the set of real numbers equipped with the standard topology. Let A and B be two subsets of…

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Q: 1 0 10 1 0 1 0 1 0 Let A = {1,2,3,4,5} and M, = 1 1 1 1 1 0 1 0 1 0 |1 0 1 0 1 a) Give the relation…

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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: A relation R is defined on Z by xRy if and only if 8 divides 3x + 5y. Prove that R is an equivalence…

A: This is a question from relation.

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: a) A = B = R and f+g: R → R is defined by (f + g)(x) = f(x) + g(x); b) A = B = R and fg: R → R is…

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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 f: A --> B and g: B --> C be maps. Show that if g of f is 1-1 and onto, then f is 1-1 and g is…

A: According to the given information,

Q: 3) Consider the subsets [0, 1] and [3,4] of R, the set of all real numbers with the euclidean…

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Q: On sets, the relation x R y given by there exists a bijective function from the set x to the set y'…

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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: Show that the o-algebra B of Borel sets contains all F,-sets.

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Q: 3) Let X be a set with > 3 elements. Show that there exist bijective functions f :X → X and g : X +…

A: Let x=1,2,3 then, f=123132g=123321 `

Q: 2. Define a relation R on Z as follows: a Rb if and only if 5 | (a²-b²). a) Prove that R is an…

A: A relation R on ℤ is defined by: a R b if and only if 5 | a2-b2 . We have to prove that R is an…

Q: If A is a set of cardinality m and B is a set of cardinality n. Prove that cartesian product A x B…

A: Introduction: Counting the number of items in a set, or its cardinality is a way to gauge its size.…

Q: Determine if the relation R on the set of all integers is reflexive, symmetric, or transitive, where…

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Q: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with the euclidean…

A: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with theeuclidean topology.…

Q: 5. Let X be the set X = {1,2,3}, and let R be the relation defined on X by x₁ Rx₂ if x1 + x2 3. (a)…

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Q: How many distinct equivalence classes exist in the relation R defined as below: x Ry + 3| (2x - )

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Q: 4) Let X be a finite set and f : X → X a map.

A: Given, X be a finite set and f : X→X  a map.

Q: Find an orthogonal set with the same span (and same number of elements) as [2] [8]} (18118)} "

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Q: Define a relation R on the set of natural numbers defined by (z, v) E R#z-yis divisible by 3. a)…

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Q: . Define a relation R on Z as follows: a Rb if and only if 5 | (a²- 6²). Find four integers which…

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Q: The closure of a compact subject of a regular space is compact.

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Q: Amongst the various possible products of four topological spaces A, B, C and D, show that (A × (B ×…

A: We shall solve this in next step

Q: a ) Let R be the relation on the set A={1,2,3,4} defined by aRb if and only if 2a>b+1. Find the…

A: let A={1,2,3,4}   aRb iff 2a>b+1   first we will find R…