Let S := a + √2 b + √2 : a, b EN Prove that S is countably infinite.

Q: Show tht if x is an integer, then gcd(x+1, x2-x+1) divides 3.

A: Given that x is an integer. To show :(a) gcd (x+1,x2-x+1) divides 3 (b) To determine for which…

Q: Let A be a non-empty set of real numbers which has a upremum and let B= {x : - x e A}. Show that B…

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Q: Assume that A and B are nonempty, bounded above, and satisfy B CACR. Show that sup B < sup A.

A: Introduction: The supremum and infimum properties are the very important properties of any subset of…

Q: a) Prove that P(EUF) ≤ P(E) + P(F).

A: Given that Prove that P(E U F) ≤ P(E) + P(F)

Q: show that Q has infinitely many distinct quadratic extensions in C

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Q: Let a and b be positive integers and let b= qa +r, where 0 ≤r <a. Prove that (a) 2b - 1 = 2" - 1…

A: Let  and  be two positive integers.Let  where (a) We  have to prove that      Now, observe that…

Q: X = { Ø₁ A₁ A², 3 and X₂ = { 0, B, B², √ ² are o fields but X, U X₂ is not. Why?

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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: 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 n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: Prove that gcd(a, b) is the least natural number representable in the form ax + by with x, y ∈ Z.

A: We have to show that gcd(a, b) is the least natural number representable in the form ax + by with x,…

Q: Prove that if C has infimum 2 then D = {x ∈ R | c ∈ C, x = 3c} has infimum 6

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Q: nd the supremum and infimum of the set D defined by D:= :neN

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Q: ak + b) {{} = m – 1). k=0

A: Please cheek the details in the next step

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: Prove: If f₁, f₂ € R(a) on [a, b] with f₁ ≤ f₂, then so f₁ da ≤ f f₂ da.

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

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

Q: Let|a| = 32, the number of right cosets of (a²) in is:

A: Given          |a| = 32 Then, &lt;a&gt; = { a , a2 , a3 ,  .  .  .  , a32 = e } ,  where e =…