7. Prove: Vn e Z, n is odd if and only if n is odd.

Q: - 21 , Prove that + (n? m' ) yn=o 2.

A: According to the Leibnitz theorem, dndxn(uv)=C0nunv0+C1nun-1v1+C2nun-2v2+... Formulae used:…

Q: 5. Prove that S(n +1, k+1) = £ $(m, k) (").

A: To prove- Sn+1,k+1=∑m=knS(m,k)nm The falling factorial is defined as the polynomial,…

Q: Let n, k, m EN with n2k21. prove that

A:

Q: prove if P>-l then Citp = I+np %3D n€ IN

A: The given statement is not TRUE. Counter example is taking n = 2.

Q: The value(s) of x that satisfy In(x² + 1) – 2 In(x) = In(16) – In(8) are/is

A:

Q: Show theit the fenias +- (AtI)(28+1) (P+1)(2p41)(38+1) (uhese a>0, B>0) conveages if B>a amd…

A: Ratio test If we have series ∑ann=1∞ Then we will find the limit l = limn→∞an+1an If l<1 , then…

Q: 5. Prove that (Z, × Z6)/((2,3)) S Z12.

A:

Q: 6. Prove that if alb and k E Z, then a|(b+ ak).

A:

Q: Suppose that > ak = a and let c E R. Prove, in all the that (c-an) = gory detail necessary, = c•Q.…

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

Q: Let S = {10n – 1: n e Z}, T be the set of odd integers, and U Prove that S CTNU. {5т + 4: т E Z}.…

A: Consider the given information. Here, S=10n-1:n∈Z And, U=5m+4:m∈Z And, the set T is the set of odd…

Q: В. Prove that A ERT ~ ASKY by SSS Similarity theorem. Statements Reasons 3 4 By Computation 15 %3D 4…

A: Given problems:-

Q: Prove both sides equal πU (PN ₂) = (π Up) N (π UV).

A:

Q: 5. Let r € R and A, = {q € Q : q < x} . Prove sup A, = x.

A: According to the given information, Consider the given set Ax:

Q: Q: Show if that IÃ+Bl= BI then Rand perpendicula are

A: Given: The sum and difference of two vectors are same in magnitude. A→+B→=A→-B→ Squaring on both…

Q: Show that for all n, m∈Z if n+m is even, then either n and m are both even or n and m are both odd.

A: Use the results of addition of integers.

Q: (3. Suppose T is a linear tronstormation from V to w and that {š X is a LI. söt of ve ctors in V.…

A:

Q: Suppose m e Z. Prove by both contrapositive and contradiction that if m² + 2m + 4 is odd, then m is…

A:

Q: 1. Prove the following in SDE. P+ Q, Q→ R, R→ S - P –→ (R A S)

A: When we infer a conclusion from a given premises in order to produce an argument, we use laws of…

Q: 11. Let SC R and suppose that s := sup S belongs to S. If u ¢ S, show that sup(SU {u}) = supfs*, 4).

A:

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

A:

Q: 7. Prove: Vn,meZ, if n and m are odd, then n·m is odd.

A:

Q: (" * ?) Prove: 1n + 2(n – 1) + 3(n – 2) + …+ (n – 1)(2) + n(1) = using lattice paths.

A: As per the question we have to prove the following summation identity : 1n + 2(n-1) + 3(n-2) + ... +…

Q: 4.51 Verify that BIBD with the parameter a = 8, r = 8, k = 4, and b = 16 does not exist.

A:

Q: Prove: Vn,meZ, if n and m are odd, then n-m is odd.

A:

Q: Let x, y E Z. Prove that if 3 ł x and 3 ł y, then 3|(2² – v²).

A:

Q: Let E Os 0s n, 0s zs 1, 8Vzs rS 8. Compute

A:

Q: 4 Let X,) be a pankizlly oadered e For x,y € X deftre xLy eff x sy and x#y. Pucose the Fallowiag:…

A:

Q: 11. Given: ZN = ZP; ZMQN = ZQMP Prove: AQPM = AMNQ Statements Reasons

A: Solution is given below:

Q: (a) Let n €N\{0}, and k e Z. Prove that k-

A: We will use very basic approach for this question in such a manner that it will enter in your head…

Q: Prove by contradiction that if m, n € Z, then m² - n is even or m-n-1 is even.

A:

Q: Prove that (S ⋂ T)c = Sc ⋃ Tc and  (S  ⋃ T)c = Sc ⋂ Tc .

A:

Q: 7) Assume dim V = n, dim U = k, dim W = n - k, and U + W = V. Prove that Un W = {0} and V = U OW.

A: Given, dim V=n, dim U=k. dim W=n-k, and U+W=V.To prove that  U∩W=0 and  V=U⊕W.

Q: discrete

A: In method of contradiction we start by assuming that given statement is false. Then prove that it is…

Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: U is a neighborhood of x0 and U ⊂ V, The V is a neighborhood of x0. (2) If U1 ...., U2 are…

A: Solution ::

Q: If tanh '() = 1- 12/3, then z = %3D In + 종 In O In In

A: As per the question we have to solve the complex equation and find the exact value of the complex…

Q: Let SCR. If sup S (resp., inf S) exists, then it is unique. ylgn ihi

A: The supremum (or least upper bound) of a set S ⊆ R which is bounded above is an upper bound b ∈ R of…

Q: 3. In the diagram below of AAGE and AOLD, ZGAE ZLOD, and AE OD. 44 To prove that AAGE and AOLD are…

A:

Q: Prove the given statement by the indicated method. (1). By Direct Proof: "Vm,n € Z*,+> 2."

A: ∀m, n∈ℤ+,We know square of an integer is always non-negative Hence (m-n)2≥0…

Q: 9. Prove or disprove. (a) Vn e Z*, n² < 2". (b) Vm e Z, if m is odd, then m² is odd.

A: In this question we have prove if this equation are valid or not.

Q: prove for n=l,2,3. e that +2+3+..+h = ncn+l) 2. %3D %3D

A:

Q: Prove or disprove the following: Vne Z, n³ + n²+1 is odd

A:

Q: 2) Let mf lu and (a,m)=| a,bE Z such that Let h= Om ( a) show and and (h,m)=1, %3D let. ne OmCab)…

A:

Q: Prove: Vn e Z,if n is odd, then 3n+5 is even.

A: Consider the provided question, To prove: if n is odd integer, then 3n + 5 is even. Proof by…

Q: show that whe ther RERiemman L-,1} yot

A:

Q: c) Compute µr, Hy, Ož, o, Cov(X, Y), and p

A: Reviewing information, Given pdf is, f(x,y)=x+y                 ; 0≤x≤1 ,  0≤y≤1

Q: Prove thet lo 0 n*-| =O Cmod 3) 24+1

A: In the given problem we have to prove by Fermat’s Theorem…

Q: 3.1 Prove: For all a and B in R, Ja + B] > | |a| – |B||. Hint: la + B] = |a – (-B)|-

A: Answer is in below steps.

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

A:

Q: what Reasow: Complete the proof that AHIK = AGJK. I HI I HJ GNew Givew Given K 26全CH Z GkJ Î CHKI ?…

A: angle G = angle H ( right angled angle) angle GKJ = angle HKI ( vertically opposite angle)…

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!