On average, how many components does a rooted forest on [n] have

Q: Prove that if G is a graph with spanning tree T and e is an edge of G that is not in T, then the…

A: Given that G is the graph with spanning tree T and e is an edge of G that is not in T Let x and y be…

Q: Suppose V is a set and that n(V) = 10. How many subsets of V contain exactly 3 members?

A: Number of ways in which 3 elements of V can be clubbed to form a subset = 10C3…

Q: Any basis set is also a spanning set. Select one: O True O False

A: Definition of basis: Set of vectors in basis are linearly independent. Also, set of vectors in…

Q: Let T be a set of subtrees of a tree T. Assume that the trees in T have pairwise non-empty…

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Q: 2. a. List all 3-combinations for the set {x1, x2, x3, X4, Xs}. Deduce the value of (,). b. List all…

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Q: Show that Loo is not icomorphic to a subepace of

A: Consider the spaces l∞   and   lp where p∈1, ∞. It is known that l∞ is not a separable space but, lp…

Q: By showing that each side is a subset of the other side, prove that A △ B = (A ∪ B) − (A ∩ B)

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Q: Let T be a tree. If u Є V(T) and v, w = N(u) with v‡w, then 2e(u) ≤ e(v) +e(w).

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Q: 3. Determine if the matrik 4 is in the span of the set S = D

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Q: The Hasse diagram of a poset, P, is given below. Find a b (i) the minimal element(s) of P, (ii) the…

A: Please find the answer in next step

Q: C) = (*,') + "- ), Give an interpretation involving subsets. Problem 4) Prove that

A: See the hand written

Q: 4. Find the Minimal Spanning Tree of the following network 35 30 N 12 18 15 39 12 16

A: Find the Minimal Spanning Tree of the following network.

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U ·. U Bn equal A exactly?…

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Q: Is Q x Z a closed subset of R2? An open subset of R²?

A: Since  is dense in , it is possible to find a sequence in which converges to . But,  not belongs to…

Q: Illustrate the key ideas in the proof of Hall's Matching Theorem.

A: A bipartite graph is a graph G = (V, E) whose vertex set V may be partitioned into two disjoint set…

Q: Which sets are compact? The union of [0, 1] and [2, 3]

A: Concepts introduction: Theorem (Heine-Borel Theorem): A subset S of R is compact if and only if S is…

Q: In the Prufer code for a certain tree T, seven numbers appear twice each, one number appears three…

A: The Prufer code helps in encoding a labelled tree having n vertices, by using a sequence of n-2…

Q: In lectures we have seen that the set {1, 2, . . . , n} has 2n subsets. Now say that a set is…

A: Find the number of set

Q: For every graph G on n vertices, either G or its complement with respect to Kn must be Eulerian…

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Q: If A and B are connected subsets of RP, give examples to show that AU B, An B, A\B can be either…

A: If A and B are connected subsets of ℝp,give example to showthat A∪B, A∩B, A/B can be either…

Q: 6. For A, B, subsets of R, define A+B= {a+b : a € A,b € B} and A·B = {ab : a € A, b E B} (a) For A =…

A: Answer: As per guidelines we can do only 3 subparts of question and the solution of this is given…

Q: If a set S has n > 3 elements, how many subsets of S have less than n – 1 elements?

A: Apply the simple logic of Permutation & Combination.

Q: d) The set E = (-1,0) U (0, 1] is an open set. %3D

A: Given set is E = (-1, 0) ∪ (0, 1]

Q: The Cantor set is formed by starting with the interval [0,1] and removing the middle third, then…

A: It is given that Cantor set is formed by starting with the interval 0,1 and removing the middle…

Q: If 1/3 of the blocks in set A are red, and 1/4 of the blocks in set B are red, does it follow that…

A: Given 1/3 of blocks in set A are red and 1/4 of blocks in set B are  red

Q: Consider the Hasse diagram of the the following poset:   a) What are the maximal element(s)? b) What…

A: Consider the given Hasse diagram of the poset.   (a) The 'K' is the only maximal element. (b) The…

Q: Fill in the blank. A distinct group of objects without regard to their arrangement is called a A…

A: topic - combination

Q: The number 52 is written as a sum of three natural numbers 52 = a + b + c (the triple (a, b, c) is…

A: We have given that a + b + c = 52 The decomposition are 14 + 15 + 23 = 52 15 + 23 +14 = 52 Above…

Q: Let S be a u,v-walk in a graph G. Show that a shortest (in number of edges) u,v-walk contained in W…

A: It is given that S be a u,v walk in a graph G.

Q: Use Venn Diagrams to show An (BAC) = (An B) A (AnC).

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Q: Let Y = {2019, 2020} and M = {CRV, Accord, Civic} %3D List the elements of Y x M.

A: Given that Let Y ={2019,2020} and M = {CRV,Accord,Civic} List the elements of YxM Explanation :- YxM…

Q: Prove that among any group of people, there are always two people with the same number of friends…

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Q: Consider X = {a1, a2, ..., an} Can a topology of X with 3 open and one with 4 open be equivalent?

A: We have to verify the given statement.

Q: Write out the elements of the set 2X where X= = {1,{0}, Ø, {2,3}}.

A: