Suppose that j and k are positive integers such that j< k. Give a combinatorial proof of t identity: k j+1, i=j

Q: 1. Let ƒ : N × N → N be defined as f(m, n) = 2m−¹(2n — 1). (a) Show that f is a bijection. (b) Using…

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Q: Find the value of k such that A is singular. 1 0 5 3 -1 0 2 k k =

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Q: Icm(a, b). Prove that Let a and b be positive integers, and let d = Za × Z, = Za × Zm· gcd(a, b) and…

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Q: (a) Prove that every element of Q/Z has finite order.

A: Note:- As per our guidelines, we can answer the first part of this problem as exactly one is not…

Q: Let p». Prove that, for every e > 0, a.a.s. the largest component of G(n, p) has size at least…

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Q: Now let ap be a nonzero element in Z/pZ. Show that a, is invertible. (Hint: Invoking the fact that a…

A: Let [a]p be a nonzero element in Z/pZ.

Q: (b) Let A¡ = [-i, 1] for every positive integer i. Find U A¡ and N1 Aj.

A: Given that Ai=-i,1 for every positive integer i. That is, A1=-1,1, A2=-2,1, A3=-3,1 and so on. To…

Q: 2. Find all values of p such that Z„[r]/{x² + 1) is a field.

A: We know that results

Q: Prove that A=I-N+N²-N³ for A & N definec as follows: N=A-I (identity), where |1 0 0 an & I= 0 1 0. 0…

A: given matrix  A=1a12a1301a23001 and I=100010001 prove  that A-1=I-N+N2-N3 where N=A-I

Q: 2) Let R be a commutative ring and a,b e R. Show by induction on n the binomial formula (a + b)" =…

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Q: Prove that if a eZ then the only positive divisor of both a and a+1 is 1.

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Q: Find the value of k such that A is singular. 1 0 3 4 -1 0 2 4 k k =

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Q: Let n > 3 where x, y , z € R", |x – y| = d > 0, and r is some positive real number. Prove that if 2r…

A: It is given that x and y are d distance apart and r&gt;d/2 where x, y, z belongs to Rn So locus z…

Q: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

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

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Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A: The objective is to find Z_2×Z_2.

Q: 2. Let R = {(x, y) : x, y E R}. Prove or disprove that R is reflexive, symmetric and/or transitive…

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Q: Prove that for every v = (x, y) E R? there exists a sequence v, = (xn, Yn) € R2, where rn, Yn are…

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Q: For allne z, gcdin,nt)=l.

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Q: Find the value(s) of k such that A is singular.

A: A matrix is singular if its determinant value is zero. So we find the determinant value of the given…

Q: 3. Find the generator of Xt = Xo + ut + oWt, where W is a standard BM, and identify the PDE…

A: 3).

Q: Prove that there cannot exist two different polynomials q,r E K, both of degree less than n. such…

A: Given K be any field and a1,a2,...,an be pairwise distinct elements of K. For each i=1,2,3,...,n, pi…

Q: If e is the unity in an integral domain D, prove that ( - e )a = - a for all a ϵ D.

A: Given that e is the unity of an integral domain D. Need to prove that (-e) a = -a.

Q: Find all distinct principal ideals of Zn for the given value of n. n = 7

A: Find all distinct principal ideals of Zn for  n=7.

Q: Let x1 V3, xn+1 = V3*" ,n > 1. Use induction to show that x, < 3 for all n E N.

A: Given x1=33,  xn+1=3xn3. Show that xn≤3 by using induction for all n∈ℕ.

Q: Theorem 1.20. Assume x 2 0. Then for every integer n 2 1 there is a finite decimal r, = ao .a¡ª2 *…

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Q: Show that for any positive integers i and j with i > j, we have T;(x)T;(x) = [Ti+;(x) +T;-j(x)].

A: we have to show, for any positive integers i and j with i&gt;j TixTjx=12Ti+jx+Ti-jx Now, we know…

Q: Prove that there exist c1, c2 ER such that for all ne N C1 E1 sin (i)<c2 =D1

A: We will use the range of the sine function to determine the real constants c1 and c2

Q: In the domain of Gaussian integers Z[i], the element 15 is 青 O Reducible O Unit O Irreducible O None…

A: Since 3∈ Z[i] and 5∈ Z[i] And 15 = 3 × 5 Hence 15 is reducible element in Z[i]

Q: 3. Let n e Z be nonzero and even. Prove that there exist unique a € {0, 1}, b e N, and odd c e N…

A: Consider the given condition,  n∈ℤis a non-zero and even. The objective is to prove that there exist…

Q: Let a be any positive number. Prove that |x| &gt;a if and only if x &gt; a or x &lt; -a.

A: Suppose

Q: Let K be the splitting field of – 5 over Q. - • (a) Show that K = Q(V5,i/3) • (b) Explicitly…

A: Hi! For the part (c), we will be needing the information that what all groups we have seen before…

Q: 3. Find the generator of X = Xo + ut + oWt, where W is a standard BM, and identify the PDE satisfied…

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Q: 4. Let R be the set of all real numbers, xER and n EN such that n > 2. If (1+x) >0, prove that Pn :…

A: Mathematical induction.

Q: Let i, r, and k be positive integers, and let s gcd(i, r). Assume that r | ik. Prove that | k.

A: To prove the divisibility property under the given conditions

Q: Let 5 be a primitive 9th root of unity in C and consider the extension Q(5) of Q. Prove that T; is a…

A: The Galois extension field is very significant in field theory. For a primitive nth root of unity,…

Q: In Z[x], let I = { f(x) E Z[x]: f(0) is an even integer). 1) Prove that I is an ideal of Z[x]. 2)…

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Q: C. Prove that neither 2 nor 17 are prime elements in Z[i] (the ring of Gaussian integers).

A: NOTE:Hi! Thank you for your question. Since,we only answer 1 question in case of multiple question,…

Q: 6. Let K be compact in R and let c E R. Prove that there exists a point a € K such that |c – a| =…

A: Given- Let K be compact in R and let c ∈ R.  To Prove- Prove that there exists a point a ∈ K such…

Q: Let A E Mnxn(F) and B = aA, with a E F. Prove by induction that det(B) = a" det(A), for all positive…

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

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Q: Suppose p > 0. Prove that there is an inner product on R2 such that the associated norm is given by…

A: For first,Let's assume that p=2, then there exists a inner product on IR2 that is given by euclidean…

Q: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

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Q: Let x and y be coprime positive integers. Prove gcd(x + y, x² + y²) E {1, 2}.

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Q: Find the value(s) of k such that A is singular.

A: to find the value of k

Q: Prove that for all k E N, k ose [coske -sink0j cose Lsinke cosko

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Q: Let a E Z, a #0. Prove that for any positive integers m, n with n<m, a2" +1 divides a2 -1. a) a even…

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