aj aj Let D be a UFD and let a = I1P and b = I1 P, where p₁,..., Pm are distinct irreducible elements of D and and b; are non-negative integers for each j = 1,...,m. Prove that a | b in D if and only if a; ≤ b, for each j = 1,..., m.

Q: Let - [8] where at least one of a or b is nonzero and set wu. Verify that ||w|| = 1. u=

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Q: Operation o additien for all (XuX), (XXanXa) E TR. Define an operation fScalar multiplication…

A: consider a set S on ℝ2 where define an operation of addition on ℝ2 by X1,Y1*X2,Y2=X1+X2+1,Y1+Y2+1…

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: 11 6 linear combinations of x, y, and z, or show that it cannot be done. Let x = 120 and z = TTTT If…

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Q: Simplify Z = [(PNR) u ā] n (PU Q) n P - R using set identities Q]

A: Set Identities:-Commutative law:- A∩B=B∩A  and  A∪B=B∪AAssociative law:-   A∩(B∩C)=(A∩B)∩C  and…

Q: Let P is integral domain and a, b, CEP. Prove if a|b and anc then c|b.

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Q: For each integer n ≥ 1, let Q be the set of all n-tuples of rational numbers. For example, Q³ ; a,…

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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: 2. For TEL(V, W), prove that T is injective if and only if 7* is surjective.

A: T is injective if and only if T* is surjective.To prove this statement, we need to show that T is…

Q: 3. Let R denote the set of real numbers. For each of the following subsets of Rx R, determine…

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Q: Let H = {0, +-4, +-8, +-12,....}. How many distinct left cosets of H in Z are there? Explain your…

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Q: ZW and ZX are supplementary and ZW and ZY are supplementary. Construct a proof that ZX = ZY.

A: <W  and < X  are supplementary angle  the sum of supplementary angles = 180 degree <W  and…

Q: Suppose A 1, .., An are sets and each f : A A i+1 is surjective. Prove or disprove: fn1° fn2 *** •…

A: To prove  the given function surjective we will use the fact that each f(i) is surjective

Q: left cosets

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

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Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

Q: Let Xis Hilbert Sface and T, SEBO), 2 EF then @+*|| = \|TI|

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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: Show that there exist m and k such that ma + kn = 1if gcd(a, n) = 1

A: Let gcda,n=1.

Q: Let a < b be real numbers, and let T Qn [a, b]. Prove that sup T = b. =

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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: Let r ERQ. Prove that for all a, b, c EQ, =cr iff a = 0 and b = c. a + br

A: Given that r∈RQ Let a,b,c∈Qa+br=cr iff a=0 and b=c

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: For allne z, gcdin,nt)=l.

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Q: Let A = {AC R| A is countable} and A2 = {A C RI A° is countable} which is a measurable space? (IR,…

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Q: Find gcd(2,3+5i) in Z[i]

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Q: If Do, D1,..., Dn... nEN, are dense open sets of reals, then the intersection D = 0 Dn is dense in…

A: Condition: Dn is the open dense sets of reals, for all n∈ℕ. To find: Intersection of infinite open…

Q: Let A = (a, a, as) and B = {b₁,b₂, b3} be bases for V, and let P = {[b₁]A, [b₂]A, [b3]A} (are)…

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Q: Prove that there do not exist a, b ∈ Z satisfying 54a − 39b = 2.

A: Solution:

Q: Let a₁,..., an EC with |aj| < 1 for all j = 1,...,n and A₁,..., An ≥ 0 real numbers satisfying 1 j =…

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Q: et D be an integral domain with unity 1. Two non-zero   elements a,b∈D are associates if and only if…

A: Let D be an integral domain with unity 1 then we have show that two non-zero elements a,b ∈ D are…

Q: Using the theorem below to construct an alternative proof to the idea that ‘being associates’ is…

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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 =…