Pascal’s Triangle You may have learned that the number of ways you can choose k different

items from a set of n items is often labelled nCk or, equivalently, Cn

k

or

n

k

.

In this question, you will prove some facts about Pascal’s triangle.

(See https://en.wikipedia.org/wiki/Pascal%27s_triangle for interesting background information.)

The only facts you may use in your proofs in this question are:

• ?k ? N, ?n ? Z

+, n = k ?

n – 1

k – 1

+

n – 1

k

=

n

k

(called Pascal’s identity)

• ?n ? N,

n

n

= 1.

• ?k, n ? Z, k < 0 ? k > n ?

n

k

= 0.

You may not use the formula

n

k

=

n!

(n – k)!k!

or any other external results that you may know.

(a) [3 marks] Prove using induction that the sum of the values in the row with index n of Pascal’s

triangle is 2n

. The first row has index 0. That is, prove:

?n ? N,

Xn

k=0

n

k

= 2n

.

(b) [3 marks] Prove using induction on n that:

?k, n ? N, n = k ?

Xn

i=k

i

k

=

n + 1

k + 1

.

Page 2/5

CSC165H1, Fall 2018 Problem Set 3

2.Binary Representation of Fractions In this question, we use the notation (x)2 to refer to

a binary representation of a number x. For example, 9 = (1001)2, where 9 is the decimal representation

of the number nine.

Let Q

(0,1) =

nm

n

| m, n ? Z

+ ? n > mo

be the set of all rational numbers between 0 and 1 (exclusive).

For any x ? Q

(0,1), its binary representation is of the form:

x = (0.b1b2b3 . . .)2

where ?i ? Z

+, bi ? {0, 1}, and x =

X8

i=1

bi

2

i

. The binary digits bi are called bits.

Note that when there are trailing zeros, one may choose not to write them. For example, 3

4

= (0.11)2.

Some rational numbers, however, may not have a finite binary representation. You are familiar with this

situation from decimal, where 1

6

= 0.16666… does not terminate.

We say that x = (0.b1b2b3 . . .)2 has a finite binary representation if and only if ?n ? Z

+, x = (0.b1b2b3 . . . bn)2,

where bi ? {0, 1}.

(a) [1 mark] What is the binary representation of 21

32

? Give the answer that uses the fewest bits. (Hint:

it has a finite binary representation.)

(b) [4 marks] Prove that 1

10

does not have a finite binary representation.

(c) [4 marks] Prove that for all x ? Q

(0,1), x has a finite binary representation if and only if x =

p

2

q

for

some positive integers p and q.

Page 3/5

CSC165H1, Fall 2018 Problem Set 3

3. [8 marks] Definitions of Asymptotic Notation. Using only the formal definitions of O, ? and T,

prove the following statements. You may not use other relationships that you have learned about O, ?

and T.

(a) [2 marks] 17n

4 – 21n

3 + 6n – 135 ? T(n

4

)

(b) [2 marks] 2

n – n

2 ? ?(2n

)

(c) [2 marks] log2 n – log10n ? ?(log12n)

(d) [2 marks] ?k ? N, nn

6? O(k

n

)

Page 4/5

CSC165H1, Fall 2018 Problem Set 3

4. Properties of Asymptotic Notation. Prove or disprove the following statements. Assume

all functions referred to in this question have domain N and range R

=0

.

(a) If f(n) ? ?(g(n)), then g(n) ? O(f(n)).

(b) [If f(n) + g(n) ? O(h(n)), then f(n) ? O(h(n)) and g(n) ? O(h(n)).

(c) If f(n) + g(n) ? ?(h(n)), then f(n) ? ?(h(n)) or g(n) ? ?(h(n)).

(d) [If f(n) + h(n) ? T(g(n) + h(n)), then f(n) ? T(g(n)).

(e) If f(n) is non-decreasing and f(n) = n

2

for odd n’s (but we do not know the values of

f(n) for even n’s), then f(n) ? T(n

2

)

Phases of Enlightenment in Buddhism Disclaimer: This work has been put together by an understudy. This isn't a case of the work composed by our expert scholarly authors. You can see tests of our expert work here. Any feelings, discoveries, ends or suggestions communicated in this material are those of the writers and don't really mirror the perspectives of UK Essays. Distributed: Mon, 26 Mar 2018 MODULE 9: The Planes of Realization (phases of illumination) Illumination is an idea and a word that is usually utilized and connected with intervention, the act of Buddhism and its advantages. More often than not, it is utilized regarding the idea of Nirvana. In most Buddhist writings and written works, illumination can be perused to have begun with the Buddha himself, along these lines the term samma-sambodhi is utilized to apply to the accomplishment of edification of bodhis or meditators going for it. This is the objective of the ways of reflection dependent on most Buddhist conventions. In the first setting, the word bodhi is a Pali and Sanskrit term that is utilized to allude to a man who has metaphorically woken up and comprehended things. It is likewise used to indicate out a particular level of comprehension or information that the Buddha has picked up when he encountered his own enlivening. This understanding gave him learning on to the regular reasons for things that adds to how human and other aware creatures come into their particular presence and furthermore the workings of the mind that adds to keeping human and other conscious creatures caught into ideas, for example, enduring, resurrection and desires. In this sense, the word bodhi would then be able to be associated with picking up a more profound level of comprehension on how a man can free him or herself from the things that makes him or her be detained on the three ideas specified previously. Edification in the Buddhist Traditions The Buddha Siddharta Gautama, is the primary recorded being to have accomplished a level of full edification. This level he has effectively accomplished is composed as sammasambuddha in the Pali dialect (samyaksambuddha in Sanskrit), or what is called as the ideal frame off Buddhahood. In the Theravada convention of Buddhism's accounts or the sutta pitaka, there are various writings and portrayals about how the Buddha has encountered arousing or edification himself. This can be found in the seventeeth part of the Vanapattha Sutta, the Majjhima. In this particular section, it is depicted how the Buddha has carried on with his life in the wilderness and how from that point, he could accomplish a condition of enlivening or edification. This, the writings appear, was done after the Buddha has effectively demolished the unsettling influences that happened in his psyche, enabling him to accomplish centralization of the brain. This has come about into him achieving the Vidhyas or the proficiencies. This incorporates the nearness of understanding into his past lives, achieving knowledge into the workings of rebirth and karma and understanding to the nearness of the Four Noble Truths. In the writings, the achievement of knowledge into the Four Noble Truths is the thing that gave illumination the other term arousing. This implies any meditator honing it has just accomplished a level of achieving a security from servitude on an incomparable nature. In addition, arousing is additionally identified with having achieved the phase of Nirvana, where sufferings are finished and the way toward being renewed never again happens. In the writings, the Buddha has asserted that the freedom he got from this is sure in light of the fact that the nearness of information transpiring has enabled him to create understanding. This have then given him certain opportunity, rendering him to be free from resurrections. In showing this idea to a meditator, the educator must pressure that the nearness of enlivening means picking up understanding into the ideas of resurrection and karma, into the nearness of the Four Noble Truths and the end of everything which add to accomplishing the province of Nirvana. Just in encountering this would freedom be surely experienced by the understudy. Accomplishing Awakening or Enlightenment The fulfillment of illumination or arousing at its fullest limit can be accomplished by a meditator by turning into a Buddha and going into Buddhahood. At the point when looked with this idea, the educator should initially disclose to the meditator the different implications and setting of the word Buddha in the Buddhist conventions. Additionally, another term Tathagata ought to likewise be disclosed to the understudy. This term signifies "the therefore gone" and is utilized as a comparable to the word Buddha. Achieving full arousing or illumination is considered in the Theravada Buddhist custom to be likened in achieving the phase of Nirvana. This implies when a meditator begins honing, the educator should set his or her way to have a definitive objective of achieving Nirvana. This is valid in other Buddhist customs also. This way includes the meditator deserting the then chains of his reality and progressing in the direction of the stopping of anguish or dukkha. This full arousing or edification is achieved by the meditator in four phases. Additionally, Budhaghosa, another specialist in the Theravada Buddhist custom has portrayed another way to achieving illumination. In the Visuddhimagga or the Path to Purification, he has portrayed what he calls the Seven Stages to Purification which depends on the Noble Eightfold Path initially depicted by the Buddha. The distinction, be that as it may, lies on the way that Buddhaghosa has underscored on understanding dependent on the three attributes of life which are dukkha, anatta and anicca. These ideas are what separates it separated from the four phases of edification where the ten shackles or human presence are relinquished in a progressive way. The Four Stages of Enlightenment In Buddhism, there are four phases of a progessing nature that is identified with the four phases of edification. These stages are critical in coming about into the full illumination of a meditator as an Arahat. The general population who are into both of the four phases of edification are alluded to by the Buddha as the ariya-puggala or the honorable individuals. On the other hand, the general population inside the network of the bikkhu-sangha are called as the ariya-sangha or the respectable sanghas. The four phases of edification are the Sotappana, Sakadagami, Anagami and the Arahat. These four phases of edification and their instructing to meditators are focal components in the Buddhist schools, for example, Theravada convention. This was chronicled in the sutta pitaka, and how each level are achieved were depicted also. Aside from the four phases made reference to over, the instructor would need to tell his or her understudies that there are additionally different composes which portrays different stages also. Be that as it may, the focal point of this module is on the four phases of edification so these four would be the ones that are to be depicted in detail here. As made reference to in the past areas, the four phases of illumination are the finished result or aftereffect of the seven cleansings that a meditator experiences. This was talked about at incredible lengths in the Visuddhimagga. The instructor can support his or her understudy meditators to peruse on passages from the Visuddhimagga about the seven cleanings, their groupings and how each of these are identified with four ways and natural products. In addition, in the Visuddhimagga, prajna and its significance is additionally portrayed in detail, and in addition picking up understanding into anatta and how these are identified with freedom and can be achieved in the act of Insight contemplation or Vipassana. The four phases of accomplishment or illumination are additionally connected with happening in sets of way and organic product. Coming up next are the way and natural product sets of the four phases of fulfillment: The way to stream section and the fulfillment of stream passage The way to once returning and the fulfillment on once returning The way to non returning and the fulfillment of non returning The way to turning into an arahant and the fulfillment of turning into an arahant Each of these are portrayed beneath in their connection to the accomplishing of edification: The Sotapanna. This is the principal phase of edification and is gotten from the Pali dialect (composed as Srotapanna in Sanskrit). This term implies or is meant mean the individual or the person who enters the streams (apadyate sota). The stream being portrayed here is the super commonplace portrayal of the Noble Eightfold Path and is viewed as the most astounding type of Dhamma also. The individual who is this stage is likewise viewed as one who could open the eye of the Dhamma (or dhammacakkhu in Pali and dharmacaksus in Sanskrit). The meditator who enters the stream is said to be capable achieve the condition of being an arahant in a range or seven resurrections after he or she has accomplished opening the eye of the Dhamma. Another angle that the instructor needs to worry in encouraging this is the meditator can accomplish a grip of the Buddhist teachings on a natural level or what is known as the correct view (samyagdrsti in Sanskrit or sammaditthi in Sanskrit). In addition, the meditator can likewise have an entire certainty, or Sadha on what is viewed as the three gems of training specifically the sangha, dharma and Buddha. This implies when the meditator passes away, he or she won't experience resurrection in any plane that is completely lower than the human plane, for example, the creature or in hellfire. The Sakadagami. This is the second phase of illumination which is additionally called the phase of the once returner. The cause of this word is Pali (Sakrdagamin in Sanskrit), which when interprets implies the once who once comes (sakrt and agacchati, separately). While clarifying what this implies, the educator should tell the understudy that individuals who have a place in this stage will return again to the human plane or world once again in many occurrences. A man who advances in this phase in the wake of experiencing the first is said to have abandone>

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