1. Write negations of: (a) Vn e Z, if n is prime, thenn= 2 or n is odd (b) 3a,beR, (ab > 0)^ (a+b<0) Write the following rational number as a fraction of two integers: 2. x = 0.2373737... Disprove: Vx, y e R, if xy >1 then x>1 and y>1 3. 4. Prove: Vn e Z, if n>2 then n -1 is composite. 5. Prove: Va,b e Z, if a|b and a|c then a|(2b – 3c)

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

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Q: 2. Let N = {A, | ne N}, where A, = {n,n+ 1, n +2, ...}. (a) Find UN and prove your conjecture. (b)…

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Q: 1), n = 1, 2, . .

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Q: 4-IN THE ADJACENT FIGURE, IF AB = AC AND DB = EC, PROVE THAT ABEC = ACBD

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Q: (b) Prove by induction that (3i – 2)² = n(6n² – 3n – 1) for n 21 %3D i=1

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Q: 12 Question Nraag Bewys die uitdrukking vir n,r EZ, 0srsn. Prove the expression for n,r E Z, 0srsn.…

A: topic - combination and permutation

Q: Prove that vn if n is divisible by 3 then n2+2 is not divisible by 3

A:

Q: 2 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction…

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Q: 5. Prove by induction that for positive integers n, E-1(3i² + 2i + 5) = ÷(2n² + 5n² + 13n).

A: Given statement is : ∑i=1n(3i2+2i+5)=12(2n3+5n2+13n) Step1: Let us check if or not the given…

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

A:

Q: 1 Prove 2.5 1 1 + .…+ 8.11 1 for n > 1. %3D 5.8 (3n-1)(3n+2)

A: Prove S(n) for n=1.   LHS =1/2.5 =1/10; RHS = 1/6+4=1/10   So it is true for n=1.

Q: Prove that E, k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

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

A:

Q: 2. Suppose A is an m x n matrir, x ɛ Rª and Ax = b. Then b is in 1) R" | 2) R™ 3) RmfnT 4) R™n

A: Given :  A is an mxn matrix , x∈Rn and Ax=b   To find :  b is in which space of the given spaces.

Q: 3. cooficiant af z" in the apransion 3)11

A: From the Binomial Expansion

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

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Q: (3) What is the order of (8,2, 5) in group Z12 × Z6 × Z7. (For this question, I don't need a strict…

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Q: Let {Xi} be i.i.d.∼ Exp(θ) and ai > 0. Denote by Z = min{a1X1, . . . , anXn}. Suppose that sum 1/ai…

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Q: 4. Let, a, = 1, az = 2, b, = 0, b2 = 1, for n 2 3, an = 2an-1+ bn-1 for n 22 bn = bn-1+ an-1 express…

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

A:

Q: 4. Let bo = 1, b1 = 3, and for cach integer k> 2, we have bz = 4bg–1 - 4bg-2 Prove that for all…

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

A:

Q: Then, we assume that Wn+1 - H Wn - l kn+1 kin 1 2 E Wn-p Wn-p p=0 p=1 (6) n+1 - € Zn - € Vn+1 = 1 Vn…

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Q: Prove by induction that E-,(i × 2' +1) = 4(n × 2" – 2" +1). 'i = 1

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Q: *3. Prove that l' + 2 + .. +n =n(n+ 1(2n + 1) for all ne N.

A: By using induction.

Q: Now, from the relations in (6) we get Wn = µ+ kn (Wn-1 + Wn-2) and zn = € + Vn (žn-1 + Zn-2). (9)…

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Q: 2-Prove that: B (m, n)-ß (a,m) 3- Simplify to the simplest: (2+0(1-0) (1+0)(3-2)

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Q: 3. By using mathematical induction proves 2' + 22 + 2 +.. +2" = 2"*! – 2 is true for all positive 3…

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Q: Prove that. 2/nJn-(n+2)Jne2 + (n+yjJney --- 3. end hence deduce that. 1xJn - (nti)Jnas -…

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

A:

Q: prove using induction: II ( 1 1 1 П 2k +1 2k+2, for n 20 (2n + 2)! k-0

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Q: Verify by mathematical induction Ž(2i-1 1. 1 - n° (2n? – 1) for all n 2 1. n(n+1) 2. Prove: V n € Z,

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Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

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Q: о Prove 2 (2n – 1) = n n - п - 1 k=1

A:

Q: Prove that Vn if n is divisible by 3 then n2+2 is not divisible by 3

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Q: 3. Prove that: na” u[n] = az (a-z)2

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

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

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Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

Q: 2- (a) Using [X, P] = ih, show that [X, PJ = 2ih& and [&, Þ³1= 2ihP. (b) Show that [, p2 = 2in(in +…

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Q: 1. Show that 2n n=0 onverges absolutely.

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Q: Show that for any neZ, n is either n= 4k n= 4ktI or n=Yk+2 n=4 K+3 or

A: Given below the detailed solution

Q: 1. For In 5n2 + 9n3 + 1, ɑn =n², prove or disprove: In O(an) (n → ∞)?

A: Given:  xn =5n2 + 9n3 + 1, αn=n2. we have to prove or disprove : xn =O(αn ) ( n→∞).

Q: 4. Prove by mathematical induction that = 2+1 1. 20+2¹+2²+...+2"

A: We have to find solutions