Let a and b belong to Q with b ≠ 0. Show that Q(√a) = Q(√b) if and only if there exists some c ∈ Q such that a = bc2.

Q: Given f(x,y, z) = In(x² + y² + z²), show that ² f , ²f , ² f x² + y2 + z2

A: Given problem:-

Q: Assume that a b•c and that gcf(a,b) = 1. Prove that a lc (hint: use the result that gcf(a,b) = 1 iff…

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Q: Prove that if a, b e Z – {0}, then gcd(a, b) = gcd(a, -b)

A: Solution:-   gcd a,b= Greatest common divisor of  a  and  b ∗Every integer  that divides  a  and  b…

Q: So then, if Az = h and Ay = V26 + h) –2 – V10 Ay Compute and reduce

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Q: Prove that K = {x+yv3 ] x, y E Z} is a subring of R.

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Q: Let A € M₂ (R) be given. (a) If AA¹ = 0n, prove that A = On. Does this hold if A € M₂(C)? (b)…

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Q: Prove that ((¬(p → r)) ≡ (r → p)) is a wff.

A: We have to solve given problem:

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: Show that f(x) = x4 + 4.x + c has at most two real roots for any umber c.

A: Consider the given equation. fx=x4+4x+c Case 1: When c>0. No sign change, by Descartes rule of…

Q: If u = x3 -3xy and v= 3x2y-y3, prove that Pu/Əx2 + Pu/əy2 = Pv/ax2 + Pv/@y²

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Q: Find an expression for sup in terms of inf E, sup E, inf F, sup F and prove it. 2F

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Q: b) If d = 1, prove that (1 if a is odd gcd(a, 2b) %3D if a is even

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Q: If m and n are positive integers and w = g(x"y"), then show that dw dw = my dy

A: Given w=g(xmyn).

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: Prove that i 224 s 2xP for xzO, y zo i and 1 t L-l with p>

A: Given that x,y≥0. And 1p+1q=1 with p>1. Consider gx=xpp+1q-x. Then g'x=xp-1-1 and g'x<0 when…

Q: Show that the function f(z) = from [0, 0) to [0, ∞) has a fixed point c. Hint: Set f(r) = z and show…

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Q: Prove that if A ∈ Mn×n(F) has two identical columns, then det(A) = 0.

A: The determinant is the scalar value computed by expanding a row or a column.

Q: Show that for all z1, Z2 E C, Then we have: |Iz1| – |z2|| < |z1 +z2l

A: 2) The inequality can be proved as follows:

Q: Let P = {p:p=6x-1, for some x EZ} and Q= {q:6y+5, for some y EZ). Prove that P= Q

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Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: U prove that In (l+x)) 2.

A:

Q: Suppose A=[v_1...v_n]. Prove that if A(x)=b is consistent if and only if b is an element of…

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Q: Prove that tr(aA + bB) =a tr(A) + btr(B) for any A, BE M11 xn(F).

A: Prove that traA+bB=a trA+b trB for any A,B∈Mn×nF

Q: Prove that X and Y are independent if and only if R = 0

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Q: Prove that if A ϵ Mnxn(F) has two identical columns, then det(A) = 0.

A: Given: A∈Mn×nF i.e. A is a matrix of order n×n.   Consider a matrix with first two column as…

Q: If A is similar to B, prove that AT is similar to BT.

A:

Q: Let n E N coprime with p. Show that there exists x, y E Z coprime and such that p | x² + ny² is and…

A: To prove the given claim we have to show that :- x,y ∈ Z ; gcd(x,y)=1 ; p | x2 + ny2   implies…

Q: Prove that det(At) = det(A) for any A ϵ M2x2(F).

A: Suppose we are given a square matrix A with four-element: a,b,c,d A=abcdAt=acbd

Q: If x > 0, prove that T - In (1+ x) > 2+2x

A: Using derivative

Q: Prove: Provided it exists, (A-1)¯ = A.

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Prove that if gcd(a, b) = 1 and gcd(a, c) = 1, then gcd(a, bc) = 1. %3D

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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: Let x ∈ R. Prove that if x 0.

A: Given :

Q: 3. If u= (x+y +z') , prove that +y

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Q: show that if Z ~ N(0, 1) and Y ~x²(n) with Z and Y to be independent, then T: n

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Q: Let P= {p:p=6x-1, for some x EZ} and Q= {q:q=6y+5, for some y €Z}. Prove that P= Q

A: Answer

Q: Prove that for the sets. B={4x+3|X€Z3 we have A =B

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Q: Find all triples of integers (x, y, z) that satisfy x² + 2y² = z2.

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Q: Find the source Q cx) Such that the So lutiim Uckit) of + Q(x) 2 ə x² 0 < x <∞ | U(x,o) = 0

A: given equation ∂u∂t=k∂2u∂x2+Qx ,       -∞&lt;x&lt;∞            ...iux,0=0 claim- find the source Qx…

Q: 6. If X has full rank, so that P = X(X'X)-1X', prove that C(P) = C(X).

A: The rank of a matrix denotes the number of independent rows or the number of independent columns in…

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

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Q: Let S = {(x, y) :x² + (y – 1)² < 1} U {(3,0)}. %3D Is the origin an extreme point of conv S? Is the…

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Q: Let a, b, c, d be real numbers where c and d are not both zero. Let X = {x E R | cx + d + 0}. The…

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Q: Prove that (aA + bB)t = aAt + bBt for any A , B E Mmxn(F) and any a,bϵ F.

A: Consider, aA+bBt=aAt+bBt    ∵A+Bt=At+Bt=aAt+bBt     ∵kAt=kAt for any scalar k.=aAt+bBt

Q: Let x, y,p E Z, where p is prime. Prove that x2 – p²y – p + 0. (Hint: you may use without proof that…

A: Solution :-

Q: Which theorem can be used to prove that ΔROM ΔΡΜΟ M P.

A: We can answer the question as below

Q: Find conditions on a, b, c, and d such that

A: Find conditions on a, b, c, and d such that  B=abcd commutes with both 1000 and 0001 For any matrix…

Q: 4. Verify thatf (x) = x³ – x2 – 20x + 6 satisfies the three conditions of Rolle's Theorem on [0,5].…

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Q: Prove that if X ⊆ Y then X ∩ Z ⊆ Y ∩ Z.

A: To prove this we will take an element from X∩Z and show that 'a' also belongs to Y∩Z.