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CM2010 Fundamentals of Computer Science, SIM, Singapore: Without using the truth table, prove if the following statements are tautologies or not
 University Singapore Institute of Management (SIM) Subject CM2010 Fundamentals of Computer Science
Posted on: 2nd Jan 2024

# CM2010 Fundamentals of Computer Science, SIM, Singapore: Without using the truth table, prove if the following statements are tautologies or not

CM2010 Fundamentals of Computer Science: Midterm Coursework

a) Without using the truth table, prove if the following statements are tautologies or not. Show your reasoning. [4 marks]
i) ๐โ(๐ โถ ๐)โยฌ๐
ii) ((Pโq) โง ((qโงr)โs)) โง (rโ(Pโs))
b) Without using the truth table, show that the following statement is true: [2 marks]
ยฌ(โx[P(x)โงQ(x)])โกโx[P(x)โยฌQ(x)]
c) Negate the following statement: [2 marks]
โ๐: ๐ท(๐) โง [โ๐: (๐ธ(๐) โง ยฌ๐(๐ฑ))]
d) For the following statement, write down a logically equivalent statement that contains no operators other than ยฌ and โจ: [2 marks]
๐ท โง (๐ธ โ ๐น)

2) Using the pigeonhole principle, prove that if we choose 14 different numbers from the following set {1, 2, 3, 4,โฆ,20}, then definitely there are two numbers such as a and b (among our 14 selected numbers) which their difference is at least 7 (i.e. |a-b| โฅ7) [3 marks]

3) Prove the following statement by induction. For all positive integers n, prove ๐๐๐ โ ๐ is divisible by 7, State the mathematical induction and show your work clearly. [3 marks]

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4) Students are required to create 6-character long passwords to access the library. The letters must be from lowercase letters or digits. Each password must contain at most two lowercase- letters and contains no repeated digits. The second letter must be always โ1โ and the last letter always must be a digit. How many valid passwords are there? You are required to show your work step-by-step. [4 marks]

Note: 1h032a is invalid because the second letter is not 1. 314g05 is valid because there is one lowercase letter, no digits are repeated, the second letter is 1 and the last letter is a digit.

5) Design a Finite State Automaton (FSA) that accepts binary numbers greater than 0, whose decimal equivalents are divisible by 5. For instance, the FSA should accept inputs like ‘1010’ (decimal equivalent: 10), and ‘101’ (decimal equivalent: 5), but it rejects inputs like ‘1101’ (decimal equivalent: 13) and ‘1001’ (decimal equivalent: 9). [4 marks]

6) Consider the following automaton:
a) Give an example of a string of length 6 containing 01 that is accepted by the following automaton. [2 marks]
b) Give an example of a string of length 5 that is rejected by the following automaton. [2 marks]
c) Describe the language of this automaton in terms of a Regular Expression. [3 marks]
d) Re-draw this FSA with at most 4 states. The new FSA should accept all strings that this FSA accepts. In addition, it should reject all strings that this FSA rejects. [2 marks]
e) Draw an FSA that accepts all binary strings that start with 0, the length of the string is even but the number of 1โs in the string is odd. For example, 0111, 01, and 000111 should be accepted while 111 and 0011 should not. [3 marks]

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