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Task Backgroundthis Assignment Involves Solving Problems By Using Var

This assignment involves solving problems by using various discrete techniques to model the problems at hand. These models develop the foundation for writing computer programming code that automates tasks. An understanding of sets, relations, graphs, finite automata structures, and grammars is essential to perform these tasks effectively.

Paper For Above instruction

Part I: Set Theory

To analyze the problem's sets in the context of a roulette wheel, we consider the following sets:

- A: the set of red numbers

- B: the set of black numbers

- C: the set of green numbers

- D: the set of even numbers

- E: the set of odd numbers

- F: the set containing numbers from 1 to 12, i.e., {1, 2, 3, ..., 12}

Based on standard roulette wheel distributions, the following set operations can be computed:

• A ∪ B: The union of red and black numbers encompasses all numbers on the wheel that are either red or black, essentially covering the entire non-green set of the roulette wheel.
• A ∩ D: The intersection of red numbers and even numbers includes all red numbers that are even.
• B ∩ C: The intersection of black numbers and green numbers is empty because green numbers are separate from red and black sets.
• C ∪ E: The union of green and odd numbers includes all green numbers plus all odd numbers from the non-green set.
• B ∩ F: The intersection of black numbers and the set {1, 2, ..., 12} includes black numbers within that range.
• E ∩ F: The intersection of odd numbers and {1, 2, ..., 12} includes all odd numbers between 1 and 12, namely {1, 3, 5, 7, 9, 11}.

Part II: Relations, Functions, and Sequences

In modeling the results of 10 roulette spins, we construct an n-ary relation in table format to depict outcomes such as number, color, and whether it's odd or even (excluding zero and double zero, which are neither). Each row represents one trial, with a primary key for unique identification. For each trial, the following data points are recorded:

• Number spun
• Color: Red, Black, Green
• Parity: Odd, Even, or Neither (for zero/zero zero)

The relation is 3-ary, as it involves three attributes per trial. Consequently, the value of n in this relation is 3.

Part III: Graphs and Trees

A decision tree models a player engaging in up to 4 bets, always betting on red, and stopping after losing twice. The tree's branches represent outcomes (win or lose), guiding the player's decisions: continue betting if they win or have fewer than two losses, stop if they have lost twice or completed four bets. Endpoints of the tree reflect the final state, whether the player wins, loses, or stops early due to losses.

Part IV: Automata Theory, Grammars, and Languages

(1) State Machine for a Subway Gate

The gate operates in two states: LOCKED and UNLOCKED, transitioning based on inputs TOKEN and PUSH. The transition rules are:

• LOCKED + TOKEN → UNLOCKED
• LOCKED + PUSH → still LOCKED
• UNLOCKED + TOKEN → still UNLOCKED
• UNLOCKED + PUSH → LOCKED

---Transition Table---

(ii) Transition Diagram

The transition diagram illustrates the state changes with directed edges labeled by the input signals, depicting the loop on each state for repeated inputs and the transition between LOCKED and UNLOCKED states based on TOKEN and PUSH inputs.

(2) Grammar for Algebraic Expressions

The context-free grammar generates algebraic expressions involving variables p, q, r, and operators +, –, ×, ÷, with rules:

1. E → p
2. E → q
3. E → r
4. E → E + E
5. E → E – E
6. E → E × E
7. E → E / E
8. E → (E)

To derive the expression (p + q) × p – r × p / (q + q), we follow these steps:

1. Start with E, and derive (p + q) × p – r × p / (q + q):
2. Using rule 8: E → (E)
3. Inside parentheses, derive p + q: E → E + E (rules 4) with E → p and E → q
4. Derive the multiplication and division parts similarly, combining rules 6 and 7 with the existing expressions
5. Complete the derivation matching the target expression step-by-step

Parse Tree

The parse tree for the expression features a root node representing subtraction, with left child associated with the multiplication of the sum (p + q) and p, and right child representing the division of r × p over (q + q). The tree visually displays the hierarchy and order of the operations as dictated by the grammar rules.

Conclusion

This comprehensive analysis demonstrates how set operations model roulette outcomes, the utility of relations in capturing repeated experiment results, the application of trees to represent game strategies, and the relevance of automata and grammars in modeling computational and language recognition tasks. Each part consolidates theoretical foundations with practical modeling approaches, essential for understanding discrete mathematics and computational logic applications.

References

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• Harary, F. (1969). Graph Theory. Addison-Wesley.
• Cohen, D. I. A. (1974). Set Theory and Its Philosophy. Dover Publications.
• Gruska, J. (1997). Foundations of Computing: Automata, Formal Languages, and Data Structures. McGraw-Hill.
• Sedgewick, R., & Wayne, K. (2011). Algorithms (4th ed.). Addison-Wesley.
• Mendelson, E. (2015). Introduction to Mathematical Logic. Chapman and Hall/CRC.
• Hopper, D. R. (1991). Coding Theory: A First Course. Springer.
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