1. Write the negation of the statements below. a. If the glass is empty, then there is no water. b. The division of two integer nonzero numbers is rational.
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- . Write a negation for each of the following statements: (a) The variable S is undeclared and the data are out of order. (b) The variable S is undeclared or the data are out of order. (c) If Al was with Bob on the first, then Al is innocent. (d) −5 ≤ x < 2 (where x is a particular real number)4. Write the negation of each of the following quantified statements. Hint: Use De Morgan's laws. a. At least one of the students submitted the homework assignment. b. Every student submitted the homework assignment or the lab assignment. c. Some student submitted the homework assignment but did not pass the exam.4. The Area of the Target: To decide on prizes for the archery contest, you decided to use the areas of the center and rings. You decided that rings with smaller areas should be worth more points. But how much more? Complete the following investigation to help you decide. a. Find the sequence of the areas of the rings, including the center. (Be careful.) b. Write a recursive formula and an explicit formula for this sequence. C. Write the explicit formula in summation notation. Find the total area of the bull's eye if it had 9 rings around the bull's eye.
- 5. We are given the statement "C3PO is a droid and Han is not a droid". (a) Using the following statement variables, write the corresponding statement form: Let p = "C3PO is a droid" and q = "Han is a droid" (b) Using De Morgan's Laws, develop the negation of the statement form of part (a) (c) What is the statement that corresponds to the statement form of part (b)?Problem 3. Provide short answers to the following questions: a. Write an example of a valid argument. b. Write an example of a sound argument. c. What is the advantage of a sound argument compared with a valid argument? d. Using the truth table for implication, show when an argument is valid but not sound.1. Let p, q, and r be the statementsp: I finish my assignment before 2:00 pm todayq: I go to the gym todayr: It is raining today (a) Invent a compound statement that uses all three of the proposi-tions p, q, and r as well as whatever logical connectives you choose. (b) Write an English sentence that corresponds to the statement youconstructed in (a). (c) Make a truth table for the statement you constructed in (a).
- (2) Let P (x, y) : x plays in y." A(x) : x is athletic." S(x) :x is smart." E(y) : y is in English league." F (y) : y is famous." Assume the domain of x is all players and the domain of y is all football teams. Answer the following questions. (a) Write the symbolic for: Some smart and athletic players play in some English league teams (b) Write the English for: 3y A(MohamedSalah) ^ E(y) ^ P (MohamedSalah, y). (c) Rewrite the statement starting with "Some... ": Not all smart players play in some famous teams3. Use De Morgan's Laws to write negations for the following statement: The train is late or my watch is fast.Problem 5 Consider the following three line argument: Some D are B. All C are D. Therefore, some D are A. Is the argument a categorical syllogism? (Q7) ? Is the argument in standard form for a categorical syllogism? (Q8) ?
- 3) There is a simple card game played with a deck of cards where a player receives four cards and two of those cards have to be same (they have to form a pair) in order to win. The computer has to determine if the player won using just one complex if statement. a) Create a systematic list of the possible combinations of cards that could form a pair (for example, Card1 & Card 2 could have the same value and make a pair). Use the table structure below. Card 1 Card 2 Card 3 Card 4 X XPlease do all given. 13 .Several logic gates can be connected together to form combinational logic. Group of answer choices True False 14. Karnaugh mapping provides a systematic method of reducing logic circuits. Group of answer choices True False There are several Boolean laws and rules that provide the means to form equivalent circuits. Group of answer choices True False- What is a term of a first-order logic formula? Please provide examples.- What is an atomic formula of first-order logic? Please provide examples.- What is a (non-atomic) first-order logic formula? Please provide examples.- Provide examples of sentences in English (math-related) that make use of “for all” and “there exists” expressions and formulate them as expressions of first-order logic.- Provide examples of FOL expressions involving the universal and existential quantifiers and translate them into English.- Discuss the following definitions associated to FOL formulas and provide examples: scope of a variable, free variable, and bound variable.