1. i. Give an example of a Boolean formula F which contains at least 2 different variables, and where F is satisfiable and its negation (not)F is not satisfiable.
ii. Give an example of a Boolean formula F which contains at least 3 different variables, and where F is satisfiable and has exactly 5 satisfying truth assignments.
i)
Satisfiable formula: (A v A') v (B v B')
Unsatisfiable formula: (A ∧ A') ∧ (B ∧ B')
ii)
Satisfiable formula: (A v A') v (B v B') v (C v C')
Unsatisfiable formula: (A ∧ A') ∧ (B ∧ B') ∧ (C ∧ C')
1. i. Give an example of a Boolean formula F which contains at least 2 different...
10 points (bonus) A propositional formula on n variables, P(ri,2,... ,Tn) is satisfiable if there exists an assignment of truth values (true or false) to its variables such that it evaluates to true. (a) Give an algorithm (pseudocode) that, given a formula P determines if it is satisfiable or not. Analyze your algorithm. b) Suppose that we are given a free" algorithm A that, given P and a partial assignment of truth values (that is, some variables are set to...
The input is a Boolean formula in 2-CNF, given as a string of symbols.Example: p /\ (p -> q) /\ (p -> ~r) /\ (~r \/ ~s) /\ (s \/ ~q)1) check whether the given 2-CNF is satisfiable.2) given a 2-CNF, report that it is not satisfiable or return one of its satisfying assignments.Solution in Python3For the first task, you should implement a function called is_satisfiable, which takes the string with the 2-CNF as an argument and returns either True (satisfiable) or False (not satisfiable). The second task...
Suppose that we add a new quantifier called exists unique to first- (d) order logic, using the symbol 3! to represent it. It means that there is exactly one element of the universe that satisfies the subsequent formula. In this question, variables will range over the universe of numbers. [1 mark] (0) Is 3!x. x + x 2 valid? Why? Give an example of a valid formula that uses the quantifier, and an example of an unsatisfiable one. Both must...
Give 2 different proofs that F is conservative. F = (ycos(x) + y, sin(x) + x) Provide at least one theorem or equation or formula number from the sheet I provided you in your solution. Make it clear where the theorem/equation is being used.
Question 1 Digital Electronics and Combinational Logic 1a) Analog and Digital Electronics i. Write either "digital" or "analog" in this to indicate whether the property in that row is - typical of digital electronics or analog electronics. The first row has been completed as an example. Property Digital/Analog Difficult, manual circuit design Analog Continuous valued signals Tolerant of electrical noise Circuit state tends to leak Intolerant of component variations ii. In older cars the timing of the electrical pulses to...
Give an example of a vector space V finitely generated over a field F , together with nonempty subsets B1, B2, and B3 of V satisfying the following conditions: (1) Each Bi is linearly independent; (2) For each 1≤I ̸= j ≤3 there exists a basis of V containing Bi ∪Bj; (3) There is no basis of V containing B1 ∪ B2 ∪ B3.
1- Given the following Boolean function in product of sums (Maxterms); F(b,c,d) = (b+c+d) (b+c+d) (b +c+d) (b+c+d) i- Use De Morgan's rule to convert Finto F ii- Complete the truth table (Table 1) iii- Simplify Fusing Boolean algebra iv- Count the number of gates in the original Fand insert them in Table.2 V- Count the number of gates in the simplified F (convert the simplified F'in part iii back to F) and insert them in Table.3 i- F' Table.1...
1. Consider the Boolean function F(x, y) = x + y, how many cells in the Kmap representing this function have value of “1”? A. 3 B. 2 C. 4 D. 1 2.Using Kmap for simplification, we can select multiple smaller groups (instead of a larger group) as long as all “1” are selected. A. False B. True 3 In Kmap representation, how many values of “0” and “1” two neighboring minterms can differ?2. Using Kmap for simplification, we can...
Please answer all questions I. Give an example of 2. List operators for arithmetic expressions in python. 3 Give an eample on expression in python 4. List operators for rational/boolean expression in python. 5. Is there any type conversion in the following expression, and why? 9/3.0