let b be a nonzero integer. prove that gcd (0,b) = |b|.

Q: Suppose x is a real number. Prove that if x 2, then there is a real number y such that x =…

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

A: Let

Q: Let H 5 {0, ±3, ±6, ±9, . . .}. Find all the left cosets of H in Z.

A: The given subset H=0, ±3, ±6, ±9,..... We have to find all the left cosets of H in ℤ.

Q: Cg + చ = చ

A: In measure theory, there is a general result known as the additivity of integrals that states that…

Q: left cosets

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Q: Show that, for all constructible numbers y € R, x³ – 2 is irreducible over • Q(r).

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: (b) Let a.. and m be positive integers. Prove that if (a,m) = (b, m) = 1, then (ab, m) = 1.

A: Given  a, b and m are positive integers.  (a, m) = (b, m) = 1.We have to prove that (ab, m) = 1.

Q: Find the infimum and supremum of the set X = {3 – Vztn ine N}. -

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Q: Let a, b eN. Prove that if a + b is even, then there exists nonnegative integers x and y such hat x²…

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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: Let f(x) = a,x" + a, p* + ...+ a, E Z[x] and a, + 0. Prove that if r and s are relatively prime…

A: It is given that,  f(x) = anxn + an-1xn-1 +...+ a0 ∈ ℤx  with an ≠ 0. Also, it is given that r and s…

Q: By using the fact that P[AU B] 1 –Ei=, P[A£].

A: (a). Prove that P [A U B U C] ≤ P[A] + P[B] + P[C]: It is given that P [A U B] ≤ P[A] + P[B]. The…

Q: Find the null space for A. = [1 null(A) A = 15-5 0 1 -³] 3 = span

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Q: Let f(x) = axm + am-prm-1 + . . . + a, and g(x) = b,x"+ b,-px"-1 + ... + bo belong to Q[x] and…

A: Given that, f(x)=amxm+am-1xm+....+a0∈Q[x]g(x)=bnxn+bn-1xn-1+....+b0∈Q[x] This implies that ai,bi∈Q.…

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 A is bounded below and B is bounded above, show that inf(A − B) = inf A − sup B

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Q: Let A and B be sets of real numbers and let B be finite. Prove that if x is an accumulation point of…

A: Let A and B be sets of real numbers and let B be finite. Prove that if x is an accumulation point of…

Q: 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…

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Q: Prove that ||A||1 = max1<j<n{£", |aij|}. %3D

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Q: Prove that R(5, 5) < 50.

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Q: define the predicate P(x) ⇔ φx(x) = 1. show that P(x) is not computable

A: Given: The predicate  is defined as , where  is the partial computable function computed by the…

Q: Let a be a non-zero element of Z43. w| 42. Show that [3] is a primitive root of Z43 and that [2] is…

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Q: (b) Let A and Bị be independent; similarly, let A and B2 be independent. Show that in this case, A…

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

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Q: find all positive integers m satisfying the condition Ø(m)=16

A: find all positive integers m satisfying the condition Ø(m)=16

Q: Which of the following is indeterminate at x = 1? x² + 1 x2 – 1 x2 – 1 Vx +3 - 2 x2 +1 Vx + 3 – 2 x…

A: We plug x=1 in each expression

Q: Determine if the following polynomials are- irreducible in

A: In the given question, the concept of Irreducible Polynomial is applied. Irreducible Polynomial An…

Q: Let x ∈ R. Prove that if x 0.

A: Given :

Q: Let x € Z. Prove that if 2 | (x²-5) then 4 | (x²-5).

A: Given that .The objective, if then prove that .

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

A: To prove that if x and y are independent then correlation between them R =0

Q: Determine if x¹ + x² + x + 1 is irreducible in Z[x].

A:

Q: Let a and b be integers which are not both zero, and let d= gcd(a, b). Prove that gcd (a/d,b/d)= 1.

A: Given that, a and b be integers and in both of them at least one is non-zero. And, greatest common…

Q: a) Factor a +x³ + x into irreducible polynomials in Z2[r];

A: i) Linear factors are irreducible. ii) Just substitute values in Zp and see the expression becomes…

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

A: Given: a and b belong to Q with b ≠ 0. To Prove: Qa=Qb if and only if there exists some c∈Q such…

Q: Prove that (z) = (zk) for every integer k (provided z 0 when k is negative).

A: