Let f (x) and g (x) be two non-zero polynomials in R [x], R being any ring. (i) If f (x) + g (x) # 0, then deg (f (x) + 8 (x)) < max (deg f (x), deg g (x)). (ii) If f (x) g (x) #0, then deg (f (x) g (x)) < deg ƒ (x) + deg g (x). (iii) If R is an integral domain, then deg (f (x) g (x)) = deg f (x) + deg g (x). %3D

Q: Prove kolmogorov's ax

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Q: 3- S Srår de

A: Find range of r and theta . Here r varies from 1 to 4 and theta varies from -π/2 to +π/2

Q: Let a = (1,2, 3), b = (1,0, –1) and c = (2, 1, 3). (a) Verify that a · (2 b+c) =2(a · b) + (a · c).

A: a=(1,2,3) , b=(1,0,-1) and c=(2,1,3)

Q: . Prove that an→L = Jan] → |L]. (Make cas

A: Given that an is a sequence which converges to L that is, an→L. Then we have to prove that |an|…

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A: We need to prove the identity  csc2z-cot2z=1

Q: Prove that {i} = I. ¡El

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Q: State the Fermet's Last Theorem. Who was French women mathematican who mede significant 19th century…

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Q: 5. Show that () = () +().

A:

Q: Show how James Garfield (1831-1881) proved Pythagorean Theorem

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Q: Prove that if gcd (a,b)=1>then gcd (ab,ab)=l

A: I have proved using the fact that gcd can be expressed as integers linear combination q

Q: How many counterexamples are needed to prove that a statement is false?

A: Suppose P(x) be the statement.

Q: State Fermat’s Theorem

A:

Q: 4. If X and Y are independent RVs, then O2+ ,(@) = ¢,(@) × ¢,(@)

A:

Q: 8. [ tan’rd.r

A: We have to find the integration of given questions.

Q: Prove that 1lellse TlI

A:

Q: 17. =- 18 6 Write an equation and solve. 18. The difference of double the number and one is…

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Q: Expend (I+x)

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Q: is R* is contractible?

A: This is a problem of Topology.

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A:

Q: Given that x, y, and z . Prove that if (x^2)|y and (y^3)|z, then (x^6)|z

A: Solution: The objective is to prove that the given statement

Q: 6- Use Newton-Raphson method to solve the system x3 +y – 1 = 0 y3 – x + 1 = 0 with the starting…

A: f1=x3+y-1=0→(1)f2=y3-x+1=0→(2)(x0, y0)=(1, 0) Formula for Newton-Raphson…

Q: 7. Theorem 94 I (9x-25)

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Q: Show that A\ (Bnc) (A\B)U(A\C).

A:

Q: How to show (0,1) is uncountable? It is from an introduction to higher mathematics.

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Q: [2 4 21 A = l1 3 1 1 1 B = -2 -1 [2 4 D = 4 C = 1 3 21 -1 %3D 2

A: Given A = 242131B = -1111-2-1C = 242131D = 1-1-1-121 Clearly  C = AD = -B

Q: Find gcd(1000, 34) and lcm(1000, 34) and verify that gcd(1000, 34) · lcm(1000, 34) = 1000 · 34

A: 1000 can be written in the form of prime factors as 1000=2×2×2×5×5×5 34 can be written in the form…

Q: 2017 J Q3 + V5 in the form a+bV5, where a and b are integers. Without using a calculator, express -2…

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Q: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

Q: .Prove that la + b + c| < |a] + |b| + ]c]

A:

Q: Show that (AUB) - (ANB) - CA-B) U (R- A) using laws of Bers.

A: The given question is solved below.

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A: Given matrix is, A=23-1160213

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Q: 1 의( (eat – ebt)} = %3D la-b (s-a)(s-b)

A: Here we have to prove that,  L1a-beat-ebt=1s-as-b

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A: