2. Let ACR. If a is a lower bound of A, then it follows that -a is an upper bound for A.

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A: We will give two proofs one based on partial sums and the other will be a direct proof.

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A: Properties of Determinants :If we given A , nxn matrix then i) The value of det(cA) = cn det(A)…

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A: Given that m,n∈ω. To prove: m+· n=m·n+n

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Q: 1. Prove that if A is infinite and a E A, then A – {a} is also infinite.

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Q: Suppose that {$1,...Øn} F p and I F ø; for all i < n. Show that I F þ.

A: Proof: Suppose ϕ∈Γ∗ϕ∈Γ∗. As Γ∗Γ∗ is maximally consistent it clearly follows that there exists a…

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

A: False.

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

Q: If E, and E2 are measurable then (E,UE,) is not measurable .

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Q: Find the least upper bound and the greatest lower bound of S = { 2pg+1 PEN, q E N pq and prove that…

A: The given set is  S = {2pq+1pq, p∈ℕ, q∈ℕ}. Now, we know that for any p∈ℕ, q∈ℕ.    pq &lt; 2pq⇒pq…

Q: If E, and E2 are measurable then (E,UE) is not measurable .

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Q: locally Lipschitz in (t,x)E[a, b]x D.

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Q: 1. Prove that if A is infinite and a e A, then A – {a} is also - infinite.

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

A: Given,

Q: Suppose T: R + P, has an action on a

A: Given, And

Q: verify that (AT )T = A

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Q: (b) Use part (a) to show that f (T) is either a transposition disjoint transpositions. or a product…

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Q: Let Sn=Vn. Find an and show that an O but that Sn=ER=1@k diverges. ->

A: Solution:-

Q: 7. (a) Find all the values of z such that e = 1+ i.

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Q: see the equation as attached here

A: Solution:Given    0&lt;an≤bn and∑n=1∞an divergesTo prove:∑n=1∞bn diverges

Q: If E, and E2 are measurable , then (E,UE,) is not measurable . صواب İhi

A: Here we have given E1 and E2 are measurable sets

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Q: Let G=(V, E) be a finite

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Q: Verify Eular,s theorem for z =ax2+2hxy+by2

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Q: 4. Let xn → 0 and (yn) be bounded. Prove xnYn → 0.

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Q: (a, o) is not measurable. ylgn ihi

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Q: Find the smallest integer in the given set a.{ xϵ Z f x &gt; 0 and x = 4s + 6t for some s, t in Z }

A: Consider the equation: x=4s+6t……1   where: x, s, t∈ℤ and x&gt;0

Q: If g(a) = a°(exin²a+eoa²a), then gʻ(a) is: atcos.

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Q: Calculate Ehe Value of:(o00001 (0000001A in GF () witlh vmc) =x°+x+X++

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Q: > a sigma algebra over X ?

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Q: If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an…

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A: For f∈L∞ , f∞=infM&gt;0:μx:fx&gt;M=0For M&gt;0, defining the set AfM=X:1 fx&gt;M⇒For f∈L∞,…

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A: Given: T∈BH and T† is an adjoint of an operator T. To show: ker T⊥=ranT†

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A: In determinant the scalar quantity is taken common from a single row or a column.

Q: . A finitely additive measure u is a measure iff it is continuous from below as in meorem 1.8c. If…

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Q: 4) Let x, y, z E Zg. Prove or disprove the following: Ifxy = xz, then y = z.

A: