8 Show that 1+2++ In S 1|+ |r2|++|rn| for any numbers r1, 22, ....

Q: 9. Show that exp(iz) = exp(iz) if and only if z = nл (n = 0, ±1, 2, ...). (Compare

A: To prove that exp(iz)¯=exp(iz¯) if and only if z=nπn=0, ±1, ±2, . . ..

Q: 4. Prove that it cannot be the case that x E AB and x EB\C. (Hint: prove via contradiction).

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: 26. In how many ways can up to 4 students be selected from 6 girls and 5 boys if each selection must…

A:

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: Prove that for every n E Z, n² – n is even. (Proof by cases)

A: In the given question we have to prove that for every n∈ℤ, n2-n is even.

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: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

A:

Q: Show that 3n^ 2+8nlogn is big-Oh (n ^2 )

A:

Q: Prove or disprove that for all n > 1 п i2' < 2"+1n. i=1

A:

Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

A:

Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: 7. For n > 1, verify that 1? + 3? + 5² + ...+ (2n – 1)? = ( 2" *1 ) %3D -

A: We have to prove that for n&gt;=1 in question number (7) Sum of squares of first n natural numbers…

Q: Prove n (2k – 1) = n² - k=1 for every n EN.

A:

Q: Prove that for every non-negative interger n: 2n+1 + (-1)" Σ 3· 2n i=0

A:

Q: IL Determine the Absolute V'alue of the following 6.-3 7-13-8) 9. 0-7 10. 9 11. 81-9 12. 43 13.-17

A:

Q: 42 Show that E a Imod n) for all integer aif and only if n is squarefree. Ep(n) is Euler Totient…

A:

Q: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

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: 1. Prove that if n E N, then E, k(k + 1) = n(n+1)(n+2). 3

A:

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

A:

Q: Show that 2"=nCk and 0 = [(-1)^nck k=0 HINT: Consider(1-1)" and (1+1)" k=0

A:

Q: or every integer n ∈ ℕ, it follows that 1^2, 2^2, 3^2, 4^2 + ...+ n^2 = (n(n+1)(2n+1)/)/6.  P

A: Claim: 12+22+32+…+n2=nn+12n+16 Proof: Now k-13=k3-3k2+3k-1 Rearranging the above equation. we get,…

Q: We need to attempt to find an exact value of n such that  P(r ≥ 6) = 0.98. Clearly to guarantee…

A: We have to fill blanks from given statement Clearly to guarantee at least six people are literate,…

Q: السؤال 2 For all n E N, n> 1, there exists a prime p such that p|n. ylan O İhi O

A: True

Q: (7) Let S = {1- (-1)" | neN}. Determine inf S and sup S. Justify. n

A: To find inf S and sup S .

Q: 5. Prove that for all n n [j³ = j=1 n² (n + 1)² 4

A:

Q: 2. Prove that n²(n + 1)? 13 + 23 +...+ n3 4. for n e N.

A:

Q: Suppose that P(A;) = 1 for all i = 1, 2, .... Show that P(N A;) = 1.

A:

Q: 1. Prove that {2" : n E Z} = {,n :n E Z}.

A:

Q: Prove that if h> -1 then 1 +nh ≤ (1 + h)” for all non negative integers n

A:

Q: Prove that when you select n +1 numbers from {1,2, ... 2n} two of them must be consecutive (that is,…

A: Use inductive argument and Pigeon hole principles

Q: Prove that (n+1)C n is equal to n+1

A:

Q: (7) Let S = {1- (-)" | n € N}. Determine inf S and sup S. Justify. n

A: To find the supremum and infimum of the sequence S .

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

A:

Q: 18. Show that Σkind(k)³ = (Σk|n d(k))² for all positive integer n.

A:

Q: Show that if M is a number less than 1, then the terms of {n/(n + 1)} eventually exceed M. That is,…

A:

Q: (5a) Let A = {3, 4} be a subset of S = {1,2,... ,6} and let n E S. || n2(n+1)² Prove that if is…

A:

Q: If nis a positive integer 1 then prove (D - a) n!

A:

Q: Prove: If |S| = n then |P(S)| = 2^n

A:

Q: 6. Prove o (k2) = k · Þ(k) for all k E N.

A:

Q: Let, a1 = 1, az = 2, b1 = 0, b2 = 1, for n 2 3, 2ап-1 + bn-1 for n 22 b, = bn–1+ an-1 an express an…

A:

Q: 15. Prove the following identity C) + (" ; ') + (" * ) + (" ; ) +-+ " : )--- " * ** ') (n + k + k…

A: Binomial Coefficient. The binomial coefficient is the number of ways of picking unordered outcomes…

Q: 4. Prove that it cannot be the case that x EA\B and x EB\C. (Hint: prove via contradiction).

A:

Q: What is the second step to prove that S, = 2+ 22 + 2' +...+ 2* = 2(2* – 1)? Show that S, is valid…

A:

Q: Prove E(2k – 1) = n² for n > 1. - k=1

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

Q: 2. Suppose that P(A;) 1 for all i = 1, 2, .... Show that P(N A¡) = 1.

A: Given problem Given that

Q: 2. Show directly that sup(4) = 1 for A = {1- : n eN}.

A: As we know that, 1&gt;1-1n ∀ n∈ℕ Therefore 1 is the upper bound of the set 1-1n:n∈ℕ. And for each…

Q: Show that 1 (N, <) is well-ordered. 1 2 (Z, <) is not well-ordered. 3 (R, <) is not well-ordered.

A: Please send questions as different questions. We cannot do more than 3 at a time.   A relation ≤ on…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.