Use induction to prove that the number of elements in any SList of depth p is 2P.

Q: Compute det (A) using the cofactor expansion along the first column.

A: First we expand using the first column

Q: By using caushy Rieman prove that C. Prove thatd az az ae

A: According to the given information, it is required to use Cauchy Riemann to prove that:

Q: Prove by structural induction that F C F2

A: Given that, F1 such that, 3∈F1, n∈F1⇒n2∈F1. and F2 such that, 3∈F2, n,m∈F2⇒nm∈F2. Consider Pn be…

Q: Prove Theorem 8.2.3 for the case where O lies in the interior of ZAPQ.

A: According to the given information, it is required to prove the given theorem for the case 0 lies in…

Q: We define the included point topology by Tp={ UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in Rif…

A:

Q: Let n, k, m EN with n2k21. prove that

A:

Q: Use the tabular method to to prove the following in general 1. (ANB) C (AUB) 2. ΑΔΒ = ΒΔΑ.

A:

Q: Suppose A ⊂ R is both closed and bounded, prove that A has the Heine-Borel Property

A:

Q: prove if P>-l then Citp = I+np %3D n€ IN

A: The given statement is not TRUE. Counter example is taking n = 2.

Q: 1. Prove by Induction that the numbr f edges of Kn

A:

Q: © Let Y Le the bunduy of the dise Dle) for ocrcl Suppise fisc on the open disc ole,n Such that Must…

A:

Q: Prove that the set S = {n² : n e N} is denumerable. %3D

A: We have to show that the given set S is a denumerable set.

Q: こ be terms Prove Find o of ů, ů. your resubts . in

A: Consider the given vectors: Here the objective is to find the vector w such that

Q: Show that the set (1) isopen u k = 1 and find its measure.

A:

Q: Show that a set {V1, ..., Vp} in R" is affinely dependent when p > n + 2.

A:

Q: prove tuat tue Complement of a d sea dense open subset Of 1R is nowhere dense in IR. N-B Please…

A:

Q: Use the result of Exercise 41 to show thatf1−α,ν1,ν2 = 1fα,ν2,ν1

A:

Q: Show that there exist m and k such that ma + kn = 1if gcd(a, n) = 1

A: Let gcda,n=1.

Q: Show that A is skew symmetric if and only if ai j = −aji for all i, j.

A: Suppose A be any square matrix of order n such that A=a11a12⋯⋯a1na21a22⋯⋯a2n⋯⋯⋯⋯⋯an1an2⋯⋯ann First,…

Q: 3. Use the above Lemma and Mathematical Induction to generalize the previous exercise. That is,…

A:

Q: Find a number t such that In(4t + 3) = 5 %3D

A:

Q: Given an n x n-matriz A over a endent iff det(A) = det(A¹,...,A") =

A:

Q: K is the midpolnt of GI and HI | GJ. Complete the proof that AHIK E AJGK.

A: Given,            K is the midpoint of GI¯ and HI¯ ∥ GJ¯

Q: In the domain of Gaussian integers Zli), the element 19 is * O Reducible O Unit Irreducible

A: Answer is irreducible. Proof is given below.

Q: Prove that all elements in S are multiples of 3. by Structualindrction 3ES X,Y ES > (X y )ES

A:

Q: Let h : N x N → N be the function h(x, y) = 3x + 2y. Use strong induction to prove that if n EN and…

A: Given that h : ℕ×ℕ→ℕ be the function h(x,y)=3x+2y. Then we have to prove by using strong induction…

Q: 10. Prove that Z has no smallest element.

A: we will prove it by contradiction, Suppose x belongs to Z  is the smallest element.

Q: If A and B is a symmeyric, proof that AB-BA is a skew symmetric?

A: We know that a symmetric matrix is a square matrix that is equal to its transpose and a…

Q: 3.) Prove that P, is a all ve ctor by showing space 10 0xioms ore

A: proof P2=a0+a1x+a2x2:a0,a1,a2∈R   is vector space over + and ×first we proof P2 is abelian group…

Q: prove that 2,

A:

Q: Prove that K4 is not outerplanar

A: In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices…

Q: is the columd of matnla A Explain why Oineanly nメn an A is invertible. independent when are 1 6

A: To explain why the columns of an n x n matrix A are linearly independent when A is invertible.

Q: show that d connot be obtained from a norm.

A:

Q: Show that K, is not planar.

A: To prove: K5 is not a planar graph. Proof:  This is proved by the method of contradiction. We use…

Q: n(n – 1) (c) Prove by induction that the number of edges of K, is (d) Sunnoca

A:

Q: Determine all the number(s) c which satisfy the conclusion of Rolles Theorem for

A:

Q: Prove that Z Q has the universal property of inverting non-zero elements.

A: in Mathematics, Z is the set of Integers. and Q is the set of all rational numbers. Z→Q Let f(α)=1α…

Q: . Let X be a r.V. such that E(X") is finite for all n = 1, 2,... Use the expansion

A: Introduction: Suppose X is a random variable. Then, the moment generating function (mgf) of X is…

Q: Use Proof by Induction to prove that 5n+1 - 1 1+5+ 52 + 53 + …. + 5" = Vn e Z, 4 Be sure to label…

A: 1+5+52+53+...+5n=5n+1-14   ∀n∈ℤ+

Q: Then ||-|| is a norm on R.

A:

Q: (b) Prove that the complement of a dense open subset of RR is nowhere dense in R.R..

A: (b)  Prove that the complement of a dense open subset of R(R) is nowhere dense in R(R).

Q: Call an L-tile to be the following shape Prove that for any nzl, a 2" With a 1 x 1 Corner Sauare…

A: For n = 1 each of the four 2×2 checkerboards with one square removed can be tiled using one right…

Q: Prove that the number V7 lies between 1 and 2.

A:

Q: Let x, y and n denote integers. Use induction to prove that Vn 2 5: 3x, y > 0 such that n= 2x + 3y…

A:

Q: Show that both ℝ and ℝ2 may be covered by countably many open balls

A:

Q: Find a polyomial f(x) of degree 3 that has the following zeros. -8, 3, 0

A: Given :  To determine the polynomial fx of degree 3 that has the following zeros : -8, 3, 0

Q: let FQ) = -2Xidefined on [14], then UC Pmif) =

A:

Q: Prove that for any k > 0, nk e o(2v™).

A: The complete solutions are given below