Let A = [0, 1] – Q. Determine if A is countable or uncountable. Prove your claim.

Q: (a, o) is not measurable.

A: to find that a,∞ is measurable or not  proceed as follows We know that every interval is measurable.…

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: By using caushy Rieman prove that C. Prove thatd az az ae

A: According to the given information, it is required to use Cauchy Riemann to prove that:

Q: find Rang, coefficiend of Rangs, Quartile devialion and cocfficient, of of the following OD classes…

A: Range= Largest term-shortest term R=L-S Here S= 1-0.5= 0.5 L= 60+0.5= 60.5 R= 60.5-0.5= 60…

Q: Let A be an mxn Aqtrix Prove that = max |Ax/ -max E l9;

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Q: Prove that the following relationship is true: nCr + nCr + 1 = n + 1Cr + 1

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Q: Prove that S4 is not isomorphic to D12.

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

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

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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: Prove that if AT A = A, then A is symmetric and A = A².

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Q: If E and E, are measurable , then (E,UE,) is not measurable. ylgn

A: False.

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: Construct a formal proof of validity for (J A K) → L.

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Q: Given an n x n-matriz A over a endent iff det(A) = det(A¹,...,A") =

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Q: Show that the Modulus Function            f : R → R, given by f(x) = | x |, is neither one-one nor…

A: Consider, f: R → R given by f(x) = | x |Where,| x | is x, if x is positive or 0| x | is – x, if x is…

Q: Let c E R and suppose that an → a. Show that can → ca.

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Q: Let A, Be M, (C) be positive definite. Prove that (4+ B)" s(A" +B").

A: Let A,B∈Mnℂ be positibve definite.  To prove A+B-1≤14A-1+B-1 Since, A and B are positive definite,…

Q: Show that [my] is o(ny)

A: In the question, it has to be xy instead of xy. So, we are solving the question by taking xy. To…

Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

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: Let s in G, s is not equal to 1.

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

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Q: Let a in F_p, a is non zero then show

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

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Q: For each subfield S of Q[i, z], determine Auts(Q[i, z]).

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Q: Show that if x is in both W and W-, then x = 0.

A: Dot product of perpendicular vectors is 0.

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

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Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: Let an= こn らnf1 a) de ter mine whetber I an] is conver g ent or diver gent b) determine whether

A: Since part a and part b both are same, I am doing part a.

Q: Prove that if A is similar to B and B is similar to C, then A is similar to C.

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Q: 2) Let a,bE Q such that a<6. Prove that ax <bx.

A: The statements can be proved by analyzing the results from the given statements. The other methods…

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: Show that For all z € C, with z + 0, then: |z|?z=1 = z

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

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Q: Let A be uncountable and B be countable. Prove that A\B is uncountable.

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Q: Let d, a, b E N. Prove that

A: Given,  d, a, b∈ℕ To prove that if d|a+b2  and   d|a-b2, then d|b.

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

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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…

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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 > 1 be an integer. Prove that n|p(a" – 1), where o is the Euler ç-function.

A: In number theory, Euler's totient function counts the positive integers up to a given integer n that…

Q: Find A' for the given choice of U. If A={positive integers}, U=Z

A: We want to find  A' for the given choice of U. If A={positive integers}, U=Z

Q: Apply any method to prove that alb (or a nand b), which is defined to be ¬(a ^ b) is complete. Write…

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Q: In Z, let H = &lt;5&gt; and K = &lt;7&gt;. Prove that Z = HK. Does Z = H X K?

A: Given: In Z H=5K=7

Q: Prove this: A=B iff A° U B = Ω =

A: Solution : First prove left to right, If A⊂B is true , All elements of A are inside B, Then Ac is…

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

A: The complete solutions are given below