Let S={1/n-1/m: m,n ∈ Z+} What are infS and supS? prove your answers. (you can use the Archimedean principle)

Q: • S/²₁(²+t+1) dt\ Evaluate (A) 0 B (D) 8 w | 00 3 ܗ ܗ 6 10 6 using the Fundamental Theorem of…

A: Given query is to find the value of integral.

Q: Show that Sn is solvable when n ≤ 4.

A: Given: Sn. We have to show that Sn is solvable when n≤4.

Q: Prove that S3 = Inn(S3) = Aut(S3).

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Q: Prove: If a > 0, then (1/a) > 0.

A: It is given that a> 0.It is known that if ab > 0 then a > 0 or b > 0 or a< 0 or b…

Q: Prove that: (5a + 6b|a, b = Z} = Z

A: To prove that .Concepts used:Consider two sets  and . To prove that , we first need to show and…

Q: [2 1] Q2) Assume A = |1 3 L3 4] [2 4 6 4] ㅋ 4 and B = 13 show that (AB)' = B' A'

A: Solution: The objective is to show that the given relation

Q: Prove that m-l dx =0 m +n (し+x)" where m,n are both Positive

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Q: 2- Let z=x+iy. Find a) Img(1/2) b) Re z c) Img (1/z')

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Q: If u * v = 2i - 3j + k, what is v * u?

A: As we know a×b = - b×a

Q: prove that l+2 +3+.. +h = n(ntl) for n=1,2g3 2 %3D

A: Given :               1 + 2 + 3 + ... + n = n(n+1)2 Let P(n) :               P(n) = 1 + 2 + 3 + ...…

Q: Is Q closed over multiplication? (Y/N)

A:

Q: S # 3 X Use the fundamental Identities to simply the expression. 1) tan & coft 2) cos²0 2 +CSCO sint…

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Q: for z = In(2® + y*). Find

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Q: re 1 da Ji rVI- (In x)²

A:

Q: Prove the following: P(n,r)=P(n-1,r)+r.,P(n-1,r-1)

A: Recall: P(n,r ) = Prn=n!(n-r)! n! = n(n-1)!

Q: Prove that bx q = b. ((Vt) - Vt), where q = V x t

A: We have to prove b→×q→=b→·∇t→Tr-∇t→ for q→=∇×t→ b→×q→=b→×∇×t→          =b→×∑i^∂∂ti×t→…

Q: Let (a, b) = 1. Prove (a2, b2) = 1.

A: Let (a, b) = 1. Prove (a2, b2) = 1. Given (a , b) = 1. Which means a is relatively prime to b

Q: Verify that ||v-w||^2=1 when ||v||=2, ||w||=1, and v*w=2

A: By the definition of inner product and norm we can derive the following formula : ‖u-v‖2=⟨u-v,u-v⟩…

Q: Show that (AB)T and BTAT are equal.

A: Now,

Q: Let A and u be positive real numbers. Then Jp(Ax) and Jp(ux) satisfy d d + dx dx (11.6.37) d ( d p²…

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Q: Let U = N, A = {1, 3, 5}, B = {x : x² - 6x + 5 = 0}, and D = {x :x > 5}. Find h. D'n (AUB)

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Q: (b) Let z = x + iy = re?". Find |z| and z² first in terms of x and y and then in terms of r and 0.…

A: Given:

Q: 2 11 Q2) Assume A = |1 3 [3 2 4 1 and B = show that (AB)' = B' A' %3D 3 6 4J 4]

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Q: Q-12)-for the model below Max z=10X1+5X2 st 4X1+2X2540 3X1+3X25 30 X12 0, X2 20 MAX Z=- O 75 O 100…

A: Linear programming. Graphical method

Q: Prove that Σ(;) = 0.

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Q: Show that al/bc, c to gives a/b ?

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Q: (a) = [ik, jl + ljk, i) (b) gim Imk] Ik

A: