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1. If your number n is odd, answer part (a) below. If it is even, answer part (b)

instead.

(a) Investigate the Sheffer stroke ↑ which represents the logical connective of alternative

denial, quoting at least two sources for the information you provide. In particular,

answer the following questions.

(i) Who was Sheffer?

(ii) When was the Sheffer stroke first described?

(iii) Is there an alternative symbol for the Sheffer stroke?

(iv) Represent each of the usual logical connectives of negation, conjunction, disjunction,

implication and equivalence in terms of alternative denial. (Show full working, and

make sure the Sheffer stroke is the only connective appearing in your final answers.)

(v) Explain the use of alternative denial in a practical application of Boolean algebra.

(Quote at least one source in your answer.)

(b) Investigate Peirce’s arrow ↓ which represents the logical connective of joint denial, quoting at least two sources for the information you provide. In particular, answer the

following questions.

(i) Who was Peirce?

(ii) When was Peirce’s arrow first described?

(iii) Is there an alternative symbol for Peirce’s arrow?

(iv) Represent each of the usual logical connectives of negation, conjunction, disjunction,

implication and equivalence in terms of joint denial. (Show full working, and make

sure Peirce’s arrow is the only connective appearing in your final answers.)

(v) Explain the use of joint denial in a practical application of Boolean algebra. (Quote

at least one source in your answer.)

(5 + 5 + 5 + 20 + 5 = 40 marks)

2. Let n be your special number. Consider the complete graph Kn on n vertices.

(a) Does Kn have a Hamilton circuit? If so, describe it. If not, explain why not.

(b) Does Kn have a non-cyclic Hamilton path? If so, describe it. If not, explain why not.

(c) Does Kn have an Euler cycle? Explain your answer.

(d) Does Kn have a non-cyclic Euler path? Explain your answer.

(4 × 5 = 20 marks)

3. Let n be your special number. Let p be the smallest prime divisor of n. Consider the complete

bipartite graph Kp,n.

(a) Does Kp,n have a Hamilton circuit? If so, describe it. If not, explain why not.

(b) Does Kp,n have a non-cyclic Hamilton path? If so, describe it. If not, explain why not.

(c) Does Kp,n have an Euler cycle? Explain your answer.

(d) Does Kp,n have a non-cyclic Euler path? Explain your answer.

(4 × 5 = 20 marks)

4. Let n be your special number. Let T be a free tree with n vertices.

(a) What is the least number of leaves that T can have? Explain your answer.

(b) What is the greatest number of leaves that T can have? Explain your answer.

(2 × 5 = 10 marks)

5. Let n be your special number. Let T be a binary rooted tree with n vertices.

(a) What is the least number of levels that T can have? Explain your answer.

(b) What is the greatest number of levels that T can have? Explain your answer.

(2 × 5 = 10 marks)

6. (a) Let n be your special number. Construct a map of a continent having n different

countries in such a way that four colours are needed to colour bordering countries in

different colours. Using blue for the ocean (and possibly for some of the countries), and

three other colours of your choice for the countries, apply colours to the map. [Use four

colours whose names start with different letters, so that you can represent each colour

with the first letter of its name.]

(b) By inserting vertices at strategic locations, convert your map into a connected planar

graph in which faces separated by an edge are differently coloured.

(c) Construct the dual graph of your connected planar graph, and apply the corresponding

vertex colouring to your dual graph.

(10 + 5 + 10 = 25 marks)

Compelling correspondence is essential to the achievement all things considered but since of the changing idea of the present working environments, successful correspondence turns out to be more troublesome, and because of the numerous impediments that will permit beneficiaries to acknowledge the plan of the sender It is restricted. Misguided judgments.In spite of the fact that correspondence inside the association is rarely completely open, numerous straightforward arrangements can be executed to advance the effect of these hindrances.

Concerning specific contextual analysis, two significant correspondence standards, correspondence channel determination and commotion are self-evident. This course presents the standards of correspondence, the act of general correspondence, and different speculations to all the more likely comprehend the correspondence exchanges experienced in regular daily existence. The standards and practices that you learn in this course give the premise to additionally learning and correspondence.

This course starts with an outline of the correspondence cycle, the method of reasoning and hypothesis. In resulting modules of the course, we will look at explicit use of relational connections in close to home and expert life. These incorporate relational correspondence, bunch correspondence and dynamic, authoritative correspondence in the work environment or relational correspondence. Rule of Business Communication In request to make correspondence viable, it is important to follow a few rules and standards. Seven of them are fundamental and applicable, and these are clear, finished, brief, obliging, right, thought to be, concrete. These standards are frequently called 7C for business correspondence. The subtleties of these correspondence standards are examined underneath: Politeness Principle: When conveying, we should build up a cordial relationship with every individual who sends data to us.

To be inviting and polite is indistinguishable, and politeness requires an insightful and amicable activity against others. Axioms are notable that gracious “pay of graciousness is the main thing to win everything”. Correspondence staff ought to consistently remember this. The accompanying standards may assist with improving courtesy:Preliminary considering correspondence with family All glad families have the mystery of progress. This achievement originates from a strong establishment of closeness and closeness. Indeed, through private correspondence these cozy family connections become all the more intently. Correspondence is the foundation of different affiliations, building solid partners of obedient devotion, improving family way of life, and assisting with accomplishing satisfaction (Gosche, p. 1). In any case, so as to keep up an amicable relationship, a few families experienced tumultuous encounters. Correspondence in the family is an intricate and alluring marvel. Correspondence between families isn’t restricted to single messages between families or verbal correspondence.

It is a unique cycle that oversees force, closeness and limits, cohesiveness and flexibility of route frameworks, and makes pictures, topics, stories, ceremonies, rules, jobs, making implications, making a feeling of family life An intelligent cycle that makes a model. This model has passed ages. Notwithstanding the view as a family and family automatic framework, one of the greatest exploration establishments in between family correspondence centers around a family correspondence model. Family correspondence model (FCP) hypothesis clarifies why families impart in their own specific manner dependent on one another ‘s psychological direction. Early FCP research established in media research is keen on how families handle broad communications data. Family correspondence was perceived as an exceptional scholastic exploration field by the National Communications Association in 1989. Family correspondence researchers were at first impacted by family research, social brain science, and relational hypothesis, before long built up the hypothesis and began research in a family framework zeroed in on a significant job. Until 2001, the primary issue of the Family Communication Research Journal, Family Communication Magazine, was given. Family correspondence is more than the field of correspondence analysts in the family. Examination on family correspondence is normally done by individuals in brain science, humanism, and family research, to give some examples models. However, as the popular family correspondence researcher Leslie Baxter stated, it is the focal point of this intelligent semantic creation measure making the grant of family correspondence special. In the field of in-home correspondence, correspondence is normally not founded on autonomous messages from one sender to one beneficiary, yet dependent on the dynamic interdependency of data shared among families It is conceptualized. The focal point of this methodology is on the shared trait of semantic development inside family frameworks. As such, producing doesn’t happen in vacuum, however it happens in a wide scope of ages and social exchange.

Standards are rules end up being followed when performing work to agree to a given objective. Hierarchical achievement relies significantly upon compelling correspondence. So as to successfully impart, it is important to follow a few standards and rules. Coming up next are rules to guarantee powerful correspondence: clearness: lucidity of data is a significant guideline of correspondence. For beneficiaries to know the message plainly, the messages ought to be sorted out in a basic language. To guarantee that beneficiaries can without much of a stretch comprehend the importance of the message, the sender needs to impart unmistakably and unhesitatingly so the beneficiary can plainly and unquestionably comprehend the data.>