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

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: (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: Suppose that E1 and E2 are measurable sets. Prove that m(E1 U E2) + m(E n E2) = m(E1) + m(E2).

A: We have to prove that the given statement.

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: 4. Assuming that the c possible subset of R (c) f(x) = V5 –

A: Use the definition of domain and range to solve this problem

Q: Prove that for all integers n > 2, f(x) = x/n is continuous on [0, 0).

A:

Q: Let S and T be subset of R. Prove the following: cl(SUT) = (cl S) U (cl T)

A:

Q: Prove that S4 is not isomorphic to D12.

A:

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: 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 ((¬(p → r)) ≡ (r → p)) is a wff.

A: We have to solve given problem:

Q: For any x ER, prove that there exists q E Z such that q-1<r <q.

A:

Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: let r ∈ R\ Q. prove that for all a,b,c ∈ Q, a+br=cr iff a=0 and b=c.

A:

Q: Prove that for all positive integers n.

A: To prove for all positive integer n ∫01(lnx)ndx=(-1)nn!

Q: Let g(x) and h(x) belong to Z[x] and let h(x) be monic. If h(x) divides g(x) in Q[x], show that h(x)…

A: Given: Let's take g(x) and h(x) belongs to Z(x) and take h(x) as monic.

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: 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 X ~ N(0, 1). For any € > 0, show that P(|X| > c) ≤ ²c - ??

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: Let A, B, C, and D be sets.

A: First, (AxC)U(BXD) ⸦(AUC) x(BUD)Take left hand side first , let  (x,y) ∈(AxC) U(BxD)=&gt; (x,y)…

Q: Prove that R is not reflexive.

A:

Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: Prove that if A, B ϵ Mnxn(F) are similar, then det{A) = det(B).

A: Given:  A, B ∈ MnxnF are similar To prove:  det(A)=det(B).

Q: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

Q: Suppose that |x| = n. Find a necessary and sufficient condition on r and s such that (x") C (x*).

A: Given,

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

A:

Q: Prove that if H ≤ G, then aHa – 1 ≤ G, for all a in G.

A:

Q: (a) Let n €N\{0}, and k e Z. Prove that k-

A: We will use very basic approach for this question in such a manner that it will enter in your head…

Q: Show that for every x E R, there exists an n EN such that <x<n.

A:

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

A:

Q: Prove that if x is not free in the formula A, then ` A ≡ (∃x)A

A: The formula is -------(1). To Prove: x is not free in the formula Proof: Let us assume that A(x) :…

Q: Let Z ~ N(0,1). Prove for all z 2 0, P(Z > x) S

A: It is an important part of statistics. It is widely used.

Q: For any measurable nonempty subset of R, L∞ (N) = I¹ (N).

A: Given: Measurable nonempty subset Ω of ℝ. To prove: For any measurable nonempty subset Ω of ℝ,…

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: Prove A = BH (x)A = (Vx)B without using the equivalence theorem.

A:

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

A:

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 for every y E R, there exists a unique x E R such that 2(x – 1)3 – y = 0. %3D

A:

Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

A:

Q: 7. Prove there exists a unique x E R such that for every y € R, ту + т — 2 3D 2у.

A: We have to find the unique value of x for which the given equation is valid. We have, xy + x -2= 2y…

Q: 11 Prove that if A CR and |A| > 0, then there exists a subset of A that is not Lebesgue measurable.

A: As per the policy, we are solving a first question. Please repost the question and specify which…

Q: True or False? If d = gcd(a,b), then there exists integers x, y such that d = ax +by

A: solution.

Q: Let o,T E Sn for some positive integer n satisfy o(k) 1< k < n. Prove that o-(k) = k and p(k) = k…

A: Solution: Let us consider the symmetric group Sn, for some positive intiger n.  We know that Sn is…

Q: Let S be some non-empty collection of real numbers. We know that Vx E S, 3y E S such that x2 y.…

A: In this type of question you need to assume the set of x,y and S that satisfy the condition given in…

Q: By algebraic properties of R, prove that for every x, y € R with y # 0 (-x) y

A: We have to prove that by algebraic property that ∀x,y∈ℝ with y≠0 -xy=-xy

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

A:

Q: Determine the ratio and interval of con (-1)"x" in=0 n6" O R.O.C=6, 1.0.C: x E [-6, 6) R.O.C=1,…

A: Here we have ∑n=0∞(-1)nxnn6n=∑n=0∞an ⇒an=(-1)nxnn6n

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

A: The complete solutions are given below