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

Q: Let a, n E N. Assume that n > 2. Show that if Va is rational, then it is an integer.

A:

Q: Prove that For all A ∈ A, we have 0≤ P(A) ≤ 1

A: Concept: Probability: Probability signifies the possibility of something happening. It's a…

Q: Let G= Z* and let * be an operation in a * b. Is a * b = ab/2 is closed under the operation?

A: Disproof by an example.

Q: If ab is algebraic over F and b ≠ 0, prove that a is algebraic over F(b).

A:

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

A:

Q: In R[x], prove ideals (x3 – 1, x7 – 1) = (x – 1, x – 1).

A:

Q: Let E be a finite extension of R. Use the fact that C is algebraicallyclosed to prove that E = C or…

A:

Q: Let a €R be algebraic over Q. Prove that Q(a) # R.

A:

Q: C. How many extensions of g to A are onto?

A:

Q: Prove that det(At) = det(A) for any A ∈ M2× 2(F).

A: Let us consider a 2*2 matrix A whose entry belongs to F, which is given below

Q: Suppose that E is an extension of F and a, b E E. If a is algebraic over F of degree m, and b is…

A:

Q: 5. Let A = {x E Z|x = 12r - 5 for some integer r} and B = {y € Z|y=6s +1 for some integer s}. Prove…

A:

Q: Prove that Q is not isomorphic to Z.

A:

Q: Let . be the operation on the cartersian product of real numbers R • R such that (a,x) * (b.y) =…

A: Given (a,x)*(b,y)=(ab,x+y)

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

A:

Q: Let a e Z. Prove that if 6 | a and 10 | a, then 15 | a.

A:

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>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…

A:

Q: Let a, b € Z with a | b. Prove that an | bn for every positive integer n.

A: Given that           a, b∈ℤ with a|b. To prove an|bn  , we shall use mathematical induction.

Q: If X has full rank, so that P = X(X'X)-'X', prove that C(P) = C(X).

A: Given information: It is given that X is a full rank matrix.

Q: Let m and n be positive integers. Prove that mZe nZ is an ideal of Z e Z and that Z eZ S Zm O Zn. mZ…

A:

Q: Let a and b belong to Sn. Prove that ßαß-1 and a are both even orboth odd.

A: Let, a and b belong to Sn.   Consider α is the product of a 2-cycles, and β is the product of b…

Q: Let a, b € Z and both nonzero. Prove that gcd(a, b) x lcm(a, b) = |ab|.

A:

Q: let theorem a, beR such that acb. Use the axioms and that : about prove a) Find R to b) Find CEIN:…

A:

Q: Prove that the ideal <x> in Q[x] is maximal.

A:

Q: a) Let R- (a+b VE: a, be Q. Prove that R is a field.

A: Since the second question is independent of the first question, as per the guidelines I am answering…

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

A:

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: Let d be an integer. Prove that Z[Vd] = {a+ bVd \a,bE Z} is an integral domain. (This exercise is…

A:

Q: To be onto, for any integer y, there exists an integer x such that y = f(x). Give a counterexample,…

A: To be onto for any real number y, there exists a real number x such that y = f(x). Let y be a real…

Q: Let a and b be any real numbers such that a 0. Prove that a < b.

A:

Q: Prove that if gcd(a, b) = 1 and cla, then gcd(b, c) = 1.

A: We need to prove If gcd(a,b)=1, then gcd(b,c)=1. The proof is given below.

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: Let ulp, d) denote the value of the Möbius function at the gcd of p and d. Prove that for every…

A:

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: 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 f, g, h : N→ R²0. Prove or disprove that if f + g € O(h), then ƒ e O(h) and g E O(h).

A: Given f,g,h:ℕ→ℝ≥0 such that f+g∈Oh. We know that f∈Oh, if h is strictly positive for all large…

Q: prove that and = nd(n)/2

A:

Q: Let a ∈ G be such that |a| = ∞ . For m≠n prove that a^m ≠a^n

A:

Q: Define H : Z → Z by the rule H(x) = 4x – 5, for all integers x. (Z: set of integers) (a) Is H…

A:

Q: 5. Let X, be a RSR (mod n). Prove that E e(E/n) EXn is a multiplicative function of n, where e(x) =…

A: Given Xn be a RSRmod n. We know that RSRmod n=x∈ℤ: 0&lt;x&lt;n, gcdx,n=1. Now consider the function…

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…

A:

Q: Let X be an infinite set. Define u : E = P(X) → Ř by %3D if A is finite µ(A) = 1 if A is infinite.…

A: Let X be an infinite set and ∑ =P(X) Let μ=P(X)=∑ → ℝ is given by μ(A)=0if A is finite1if A is…

Q: Let a, b ∈ Z, not both zero. Prove gcd(−a, b) = gcd(a,−b).

A: We have gcd(-a , b)=gcd(a,-b) let gcd(-a , b)=d Where -a=de  , b=df  ⇒a=d(-e) , -b=d(-f)⇒d is…

Q: Let A ∈ Mn×n(F). Under what conditions is det(−A) = det(A)?

A: In determinant the scalar quantity is taken common from a single row or a column.

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

A:

Q: Let Rt st a a, b€ Z and let D=R a-4. 9. show that o E homomosphic Romemerphic Momor

A:

Q: Va, beZatb = a + b +2 and aob = a tabtb is a Ring!

A: