If a+a e c(R) for all ae R. then Commutative is prove that

Q: Prove that for all a € R, 0 < r² + 2 VI

A: Prove the following

Q: Prove that 1/r + 1/r2 + 1/r3 + . . . = 1/r − 1, for |r|  &gt; 1.

A: we have to prove that: 1r+1r2+1r3+. . . . . .=1r-1  ,  for r&gt;1

Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

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

A:

Q: Let x, y E R. Prove that if 21x + 35y = 45, then x and y are not both integers.

A:

Q: Complete the congruence statement. AJKL =A_ by J K M.

A: Congruency of Triangles: Two triangles are said to be congruent if they follow the following…

Q: Let A = {x ∈ Z | 6 divides x} and B = {x ∈ Z | 3 divides x}. Prove that A ⊆ B, and then prove that…

A: We can prove using like showing every element in one set is an element of another set like that.

Q: Prove that if |z| = 3, then %3D 5 2z – 1 7 - 13 4+ 22

A: The given is related to complex numbers. The solution is explained below.

Q: Prove that 1/ r + 1/ r2 + 1/ r3 + . . . = 1/ r−1 , for ∣r∣ &gt; 1.

A: Consider, It is a geometric series with,

Q: Prove that the operation * defined on M2x2(R) by taking A * B = A. B V A,B E M2x2(R). %3D is an…

A: Given that A*B=A.B ∀ A, B∈M2×2(R)

Q: prove that: DO e de a e +6 2/3 cwhere a, b > o

A: Where a, b&gt;0

Q: HeW. By the Same way Prove that: X Ju (x) (X)

A:

Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

A:

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

A:

Q: let R be for of King and acR.UThen a3-a Prove all that R is commutative

A:

Q: - Prove the following: gcd(m,n) = 1 + 3x, y E Z such that mx + ny = 1. One direction can be done by…

A: We will prove this by using some properties of division. One way will be done with the help of…

Q: Prove that ifx > 3 and y> 4, then xy > 11.

A:

Q: Part Ca):- a*t a e CCR) for all ae R sthen prove that R is commutative.

A:

Q: If A and B are independent, prove that B and the complement of A are independent.

A: Given,A and B are independent events , so P(A∩B)=P(A)×P(B) .

Q: Suppose that A ¥BSCN D and x E A. Use the method of "proof by contradiction" to show that if x e C,…

A: Given data : x∈Aandx∉C To show : x∈B

Q: 1 Show that n2 + 1 IN V

A:

Q: Prove the following. 2. If alc and a+ b = c, then alb.

A:

Q: Prove that if Tr(AT A) = 0, then A = O .

A: Proof:- Suppose A is a square matrix of order n, That is, A=aij For 1≤i,j≤n Now we need to prove…

Q: Prove than any pair of elements in H can be mapped to the positive y-axis by an element of SL(2, R).

A: According to the given information, it is required to prove that any pair of elements in H can be…

Q: (a) Prove that de 2n a>b>0 Jo a+ bcose Ja? - b2

A: We will use basic knowledge of Integral Calculus and trigonometric identities to solve this…

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

A:

Q: If gcd(a, c) = 1 and gcd(b, c) = 1, prove that gcd(ab, c) = 1.

A:

Q: Let x ∈ R. Prove that if x 0.

A: Given :

Q: p) Prove that if ac | bc and c +0 then a | b.

A:

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

A:

Q: Prove that: If f E Ra, b), then |fI E Rla, b) with

A:

Q: Let z E C. Prove that if 1+z +...+z"-1 – nz" = 0, then |2| <1.

A: Given that, z∈C and, 1+z+...+zn-1-nzn=0 It is known that, 1+z+z2+z3+....+zn-1=1(rn-2-1)(r-1) So the…

Q: Prove that R (D-1 and R)%=(-)

A: The objective is to prove that  Pn1=1Pn-1=-1n

Q: 2. Let A € M, (R) be given. (a) If AA = 0n, prove that A = 0₁. Does this hold if A € M₂, (C)? (b)…

A:

Q: Let a * b = 2ab be defined on Z*. 1. Is * Commutative? 2. Is * Associative? 3. What is the identity…

A:

Q: ,,E R, nxn mathix, Such that For nZ 2 and any X,,X2, if P dis an nxn mattx such thet P dis (P);;…

A: We have to prove the given statement by induction.

Q: If X ∼ N (μX , σ2X) and Y ∼ N (μY , σ2Y ) are independent, then prove

A: X~Nμx,σ2x and Y~Nμy,σ2y are independant

Q: prove that for operatrs a and b which comute with thier commuters[a,b]=c,so that [a,c}=[b,c}=0 ,…

A: We know that ekABe-kA=B+kA,B+k22!A,A,B+... Given that A and B commute with their commutator then…

Q: By using Caushy Rieman prove that (2) and (3) (2) 4ca) = e (3) .w prove Thot()-

A: (a)

Q: Suppose AnB CC\D and x E A. Prove that if x eD then x ¢ B. (Hint: prove via contrapositive)

A:

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

A: We can answer the question as below

Q: Under what conditions on A and B is (AB)-1 = A-1B-1? Prove your assertion

A: Assertion: If A and B are commutative, then  AB-1= A-1B-1.

Q: Suppose that B is a point between A and C. Prove that AB−→ =AC−→

A: B is a point on AC. Let, A=(x1,y1) and C=(x3,y3)Then direction vector of AC=(x3 - x1, y3-y1)