3. Find the number of elements of the set P= {n|n E Z, 1

Q: Give an enumeration procedure for the set L = {ab*a™: n> 1, m >1}.

A:

Q: 2. Use an edge-counting argument to show that K, cannot be drawn on a 2-holed torus without edges…

A: Suppose K9 had a 2-cell embedding in the torus. Then, this embedding has v=9 vertices and…

Q: 2. Let A Ø. If A has exactly n elements, then A has exactly 2" subsets, Vn € N.

A: A is non empty subset of n elements ,then A has exactly 2n subsets,∀n∈ℕ.

Q: 5) now that for all n EN, if A1,A2, ..., A and B are any sets, then n 11 nc₁x) = (N₁) va (A₁\B) B.…

A:

Q: 2. Let S be the set [9] = {1.2..9}. 1 (D) How many derangements of S have 9 in the 2nd position and…

A: We have the set 9=1,2,...,9 We need to find the number of derangements. There are 9 places to fill…

Q: Let A and (A, n2 1) be subsets of 2 An - A as n → 00 =P(A,) P(A). as n 0o,

A: The given question is related with probability. Given that A and An , n≥1 are subsets of Ω . We have…

Q: 8. Sketch the sets X = {(x, y) e R :x+y s1} and Y = {(x,y) ER:-1sys0} on R. On separate drawings,…

A: And,

Q: Recall that P(n, r) is the number of r-permutations of an n-element set. Prove the following…

A:

Q: Show that for n, m e Z+, gcd(2" – 1,2 – 1) = 28cd(n,m) – 1.

A: To prove : gcd(2n-1, 2m-1)=2gcd(n,m)-1where n,m ∈ℤ+

Q: 3. (a) For each n e N, let An = {nk : k E N} %3D i. Determine A1N A2 ii. Determine U An n=1 iii.…

A:

Q: What is the largest value of r for which N(2.2, r) is a subset of S = ( – 5, 3) U (1.7, 4.6) .

A:

Q: 4. 5. For n E N, { x ER | |x-x*|<1/n } is open. UA_n is open where A_n=(n-1,n).

A: To determine: If the given statements are always true, sometimes true, or never true: (4.) For n∈ℕ,…

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: 2. Enumerate (i.e. list) the members of the following sets: a. {z |z is odd positive integer and z <…

A:

Q: 2.29. Let A = [a1, a2] and B = [b, b] be closed and bounded intervals in R. In each of the following…

A: Let A=[a1,a2] and B=[b1,b2] be closed and bounded intervals in R. In each of the following groups…

Q: Show that the following subset of R is countable: {m +n/2: m, n E Z}.

A: Fairly simple Z is countable set and as so set is , so now clearly we have a bijection    and that…

Q: Show that if P(A;) = 1 for all i > 1, then P(N1 A¡) = 1.

A: We know that P(Ai) = 1 for i ≥1 Considering P(Ai), i=1,2,3,... to be independent events

Q: Let X, Y subset U. Suppose that n(X)= 22, n(Y)= 20, and n(X intersect Y)= 16. How many elements are…

A: Let X,Y be subsets of U.  nX=22 , nY=20 and nX∩Y=16 It is given that nU=40. Find nX'∪Y:

Q: Show that for every x E IR, there exists an n EN such that <x<n.

A:

Q: (c) Show that a permutation o E S, commutes with r if and only if the permu- tation f(o) commutes…

A:

Q: Suppose that N+ 1 balls, of which a+b+1 are red and N-a-bare blue, are arranged in a linear order in…

A: From the given information, There are N+1 balls, of which a+b+1 are red and N-a-b are blue arranged…

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: 1. Find the union and intersection of each of the following families: (i) M ={ n Z:nEN}, where n Z =…

A:

Q: et X ∼ Bin (n,π), and Y ∼Bin (n, 1−π). Show that P(X ≤ k) = 1 − P(Y ≤ n − k − 1)

A: PY≤n-k-1=Cn-k-1nπk+11-πn-k-1+Cn-k-2nπk+21-πn-k-2+........+C0nπn1-π0 Note that Cxn=Cn-xn. Hence we…

Q: Let A = {2n : n e N}, and B = {n E Z : n² is even}. Neither set is a subset of the other. O AC B. O…

A:

Q: 2. Compute the convolution of the following signals for n 20 n n x1(n) = and x2 (n) = (;)" u(n) n…

A: Consider the expression of . Determine the z-transform of .

Q: Finish labeling the number of elements in the regions in the Venn diagram shown to the right, where…

A: We will find the all values as shown below by using given information

Q: Prove that the following 2 sets are equal: {4k + 5 : k ∈ Z} = {4k + 1 : k ∈ Z}.

A: Two sets are equal if and only if They have the same element  We are using here if x €A. &lt;=&gt;…

Q: Use the set-roster notation to indicate the elements in each of the following sets. a. S- {n € Z n=…

A: Hey, since there are multiple subparts posted, we will answer first 3 questions. If you want any…

Q: d) The set E = (-1,0) U (0, 1] is an open set. %3D

A: Given set is E = (-1, 0) ∪ (0, 1]

Q: 19. Show that (for n = 0, 1, 2,..) (a) F(-n, b; c; 1) = (c-b)n (c)₁

A: We have to show F-n,b;c;1=c-bncn Solution: By Pochhammer definition cn=Γc+nΓ(c)           (1) and…

Q: Prove, using an element-chasing argument, that {15n : n E Z} = {3n : n E Z}N{5n : n E Z}.

A:

Q: Let S = {(x, y) E R X R: y - x = integer} List of four members of [-3]

A: GivenS=x,y∈IR×IR: y-x=Integernow-3=x∈IR: -3-x=Integernow, 1∈IR,2∈IR , 3∈IR and 4∈IRso the four…

Q: Finish labeling the number of elements in the regions in the Venn diagram shown to the right, where…

A: Total number of elements in universal set, n(U)=100 Total number of elements in subset A, n(A)=30…

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

A:

Q: (3.2) Indicate which of the following sets are compact in X. (a) C = {4,n € N}, X = R (b) C = Qn (0,…

A:

Q: Let T = > Σ - In k. n n=1

A: We will solve here only (a) part of 2.   we have to show that the sequence Tk is monotone…

Q: w that the set S = 3 6. 4 5 ms all of R³ or find an element of R³ not belonging to span(S).

A:

Q: (b) Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7. Deduce that…

A: Given X, Y and Z are subsets of 1, 2, 3,.......,10. Cardinality X=Y=Z=7. To prove that X ∩ Y ∩ Z is…

Q: .....………., ɑn are in A.P., where a; > 0 for all i, show that : 1 If a1, a2, a3, 1 1 n- 1 %3D Va1 +…

A:

Q: Show that the number of planted plane trees on n vertices, m of which are leaves (that is, have zero…

A: Solution

Q: Suppose that X is a subset of Y, such that P(Y)>0. Is true or false that P(X|Y) is equal to…

A: It is given that X is a subset of Y This means that PX∩Y=PX...(i) The formula for PX|Y is given by:…

Q: Define the sets D = In E N} n+1 Prove whether D is finite, countable, or uncountable?

A:

Q: Compute 0.5x for the following set of n = 7 values. x1 = 4, x2 = 19, x3 = 18, x4 = 10, x5 = 16, x6 =…

A: Solution: From the given information, n=7 and the set of values are

Q: Suppose XX is a set with 144 elements that is partitioned into four subsets: X1, X2, X3, and X4. If…

A: To determine the number of elements in each subset in the partition.

Q: C. Let S = {(x, y) ERX R: y - x = integer} List of four members of [-3]

A: we need to find (x,y) such that  y-x=-3 where x∈R and y ∈R so some of the possible values of the…

Q: If D is a subset of R and c is a cluster point of D, implies which of the following? (a) N*(c, 1/n)…

A: Let D be a set and c is a cluster point of D. Then we have to check above statement implies which of…

Q: If X(z) = In(z - 2), then what's x(k O a. -ak O b. ak O c. -ak /k O d. others Oe. ak / k O O

A: Given that X(z) = ln(z - 2) We have to find the value of X(k).

Q: Compute (x – 2) for the following set of n = 7 values. X1 = 8, x, = 15, x3 = 14, x4 = 10, x5 12, x6…

A: Given,   =?

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!