1. Which of the following statements are true? I. If A and B are independent then A' and B are independent. II. Pr(B|A)+ Pr(B'|A) = 1 Pr(B|A)+ Pr(B|A') = 1 Pr(B|A)Pr(A)+ Pr(B|A')Pr(A') = 1 III. IV.

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A: This can be solved as follows:

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A: LetS⊆R, claim: closure(S)=S∪Bd S =S ∪ S' by definition bd(S) ={ x∈ℝ/x∉S° &amp; x∉(ℝ-S)°} let  x∈S∪S'…

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A: The symmetric group S3 is given as S3=I, 1 2, 1 3, 2 3, 1 2 3 1 3 2 where  I=123123, 1 2=123213, 1…

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A: See the attachment

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A: The objective is to prove that r≡S≡r↔S is a tautology.

Q: now if we take F (x, y) = (ax, ay), where a is any positive constan %3D

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A: Since the question has multiple sub parts we will solve the first three sub parts. Please resend the…

Q: 2] Let x,yER Prove that VxVy(10x+21y>21⇒(x>11/10 v y>10/21))

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A: The given statement means the following.

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A: Given that             fz=z4-5z2      and    C=z∈ℂ : z=1.

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A: # using Jacobian transformation we have proved that y=tanx follow Cauchy distribution (0, 1)

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A: In this question we use congruence modulo.The following  are useful in this question1) If  3 |…

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A: We have to find derivative

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A: The integration of the integral is tanhu.

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A: Since you have asked for a multiple question, we will solve the first question for you. If you want…

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A: Since you have asked multiple questions , we will solve first question for you. If you want any…

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A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

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A: This question is related to integration using gamma function.

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A: Using the substitution ... Explanation of the solution is given below...

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A: This is a problem of Number Theory.

Q: Q3. Let z, y be real numbers such that ry 1 and z #y. Prove that

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Q: Which of the following is a general solution to the followin x²y"+xy'+ (x²– 1)y=0 y=c,(x) +cY_,(x) O…

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