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(7) Write carefully the (very short) proof by contradiction of the proposition "Ifr&Q (that is, r...
25. (2 points) Below is a proof presented as a proof by contradiction. Restate the proof,...
25. (2 points) Below is a proof presented as a proof by contradiction. Restate the proof, using the same ideas, as a proof of the contrapositive of the proposition. Proposition: The sum of a rational number and an irrational number is irrational. Proof: Suppose BWOC that there existr e Q and neR-Q such that run e Q. Sincer is rational, r = for some p, q E Z. Sincer+ne Q, also r+n= for some a, b e Z. Now: r...
6 and 7 Question Completion Status: QUESTION 6 Determine whether the following compound proposition is a...
6 and 7
Question Completion Status: QUESTION 6 Determine whether the following compound proposition is a tautology, a contradiction, or a contingency. Ilo )(q )] + (0 ) o O A. All of the above OB. Tautology C. Contradiction D. Contingency QUESTION 7 Using the truth table determine if the following proposition is a tautology, a contradiction, or a contingency. [(p ) Ap] Tautology Contingency Contradiction None of the above QUESTIONS Fill out the truth table and decide if the...
(40 pts) Consider the transitivity of the biconditional: ((P HQ ) ^ ( Q R ))...
(40 pts) Consider the transitivity of the biconditional: ((P HQ ) ^ ( Q R )) → ( P R ). a. Show that this argument is valid by deriving a tautology from it, a la section 2.1. You may use logical equivalences (p. 35), the definition of biconditional, and the transitivity of conditional: (( P Q) ^ (Q→ R)) → (P + R). Show that this argument is valid using a truth table. Please circle the critical lines. C....
In this assignment you will write code that will prove both equations for three logical equivalences...
In this assignment you will write code that will prove both equations for three logical equivalences (pick any three except the double negative law). Below is the list of logical equivalences. Please create a program that allows a user to test logical equivalences and have proof of their equivalency for the user. The rubric is below. Submit screen shots of the code, input, and output of the program. Theorem 2.1.1 Logical Equivalences Given any statement variables p, q, and r,...
4:12 7 11 Exit D Question 7 15 pts Let p, q be the statements: p:A...
4:12 7 11 Exit D Question 7 15 pts Let p, q be the statements: p:A person is a clown q:A person wears makeup a) Write the following argument in symbolic form: All clowns wear makeup. Pennywise wears makeup Therefore, Pennywise is a clown b) Identify by name the argument in part a. c) Is the argument in part a valid or invalid? 12pt v Paragraph B IU ! p MO O words
QUESTION 3 Symbolize the following argument using the variables p, q, and r. Then construct a...
QUESTION 3 Symbolize the following argument using the variables p, q, and r. Then construct a complete truth table to show whether or not the argument is valid. Use 1 for T(true) and 0 for F(false). Valid or Invalid? Why? Prove. Explain what your truth table shows. 10 points Total: 3 points for correct symbolic form, 4 points for valid/invalid and reason, 3 points for correct truth table. If Max studies hard, then Max gets an 'A' or Max gets...
4. Consider the points P(1,0,1), Q(-2, 1, 4), R(7,2, 7). (a) Write the linear equation for...
4. Consider the points P(1,0,1), Q(-2, 1, 4), R(7,2, 7). (a) Write the linear equation for the plane through P, Q, and R. (b) Write parametric equations for the line passing through P and Q.
2. Using the given logic rules, write a proof sequence for the following assertion. (11 points)...
2. Using the given logic rules, write a proof sequence for the following assertion. (11 points) pva 4 → pist st זר 74 → Us Statements 1. p Va 2.91 3.(p Ast 4. r 5.9 → (UA) 6. Reasons Given Given Given Given Given 7. 8. 9. 10. 11. t
#7. TRUE/FALSE. Determine the truth value of each sentence (no explanation required). ________(a) k in Z k2 + 9 = 0....
#7. TRUE/FALSE. Determine the truth value of each sentence (no explanation required). ________(a) k in Z k2 + 9 = 0. ________(b) m, n in N, 5m 2n is in N. ________(c) x in R, if |x − 2| < 3, then |x| < 5. #8. For each statement, (i) write the statement in logical form with appropriate variables and quantifiers, (ii) write the negation in logical form, and (iii) write the negation in a clearly worded unambiguous English sentence....
Consider the following: Algorithm 1 Smallest (A,q,r) Precondition: A[ q, ... , r] is an array...
Consider the following: Algorithm 1 Smallest (A,q,r) Precondition: A[ q, ... , r] is an array of integers q ≤ r and q,r ∈ N. Postcondition: Returns the smallest element of A[q, ... , r]. 1: function Smallest (A , q , r) 2: if q = r then 3: return A[q] 4: else 5: mid <--- [q+r/2] 6: return min (Smallest(A, q, mid), Smallest (A, mid + 1, r)) 7: end if 8: end function (a) Write a recurrence...
25. (2 points) Below is a proof presented as a proof by contradiction. Restate the proof, using the same ideas, as a proof of the contrapositive of the proposition. Proposition: The sum of a rational number and an irrational number is irrational. Proof: Suppose BWOC that there existr e Q and neR-Q such that run e Q. Sincer is rational, r = for some p, q E Z. Sincer+ne Q, also r+n= for some a, b e Z. Now: r...
6 and 7
Question Completion Status: QUESTION 6 Determine whether the following compound proposition is a tautology, a contradiction, or a contingency. Ilo )(q )] + (0 ) o O A. All of the above OB. Tautology C. Contradiction D. Contingency QUESTION 7 Using the truth table determine if the following proposition is a tautology, a contradiction, or a contingency. [(p ) Ap] Tautology Contingency Contradiction None of the above QUESTIONS Fill out the truth table and decide if the...
(40 pts) Consider the transitivity of the biconditional: ((P HQ ) ^ ( Q R )) → ( P R ). a. Show that this argument is valid by deriving a tautology from it, a la section 2.1. You may use logical equivalences (p. 35), the definition of biconditional, and the transitivity of conditional: (( P Q) ^ (Q→ R)) → (P + R). Show that this argument is valid using a truth table. Please circle the critical lines. C....
In this assignment you will write code that will prove both equations for three logical equivalences (pick any three except the double negative law). Below is the list of logical equivalences. Please create a program that allows a user to test logical equivalences and have proof of their equivalency for the user. The rubric is below. Submit screen shots of the code, input, and output of the program. Theorem 2.1.1 Logical Equivalences Given any statement variables p, q, and r,...
4:12 7 11 Exit D Question 7 15 pts Let p, q be the statements: p:A person is a clown q:A person wears makeup a) Write the following argument in symbolic form: All clowns wear makeup. Pennywise wears makeup Therefore, Pennywise is a clown b) Identify by name the argument in part a. c) Is the argument in part a valid or invalid? 12pt v Paragraph B IU ! p MO O words
QUESTION 3 Symbolize the following argument using the variables p, q, and r. Then construct a complete truth table to show whether or not the argument is valid. Use 1 for T(true) and 0 for F(false). Valid or Invalid? Why? Prove. Explain what your truth table shows. 10 points Total: 3 points for correct symbolic form, 4 points for valid/invalid and reason, 3 points for correct truth table. If Max studies hard, then Max gets an 'A' or Max gets...
4. Consider the points P(1,0,1), Q(-2, 1, 4), R(7,2, 7). (a) Write the linear equation for the plane through P, Q, and R. (b) Write parametric equations for the line passing through P and Q.
2. Using the given logic rules, write a proof sequence for the following assertion. (11 points) pva 4 → pist st זר 74 → Us Statements 1. p Va 2.91 3.(p Ast 4. r 5.9 → (UA) 6. Reasons Given Given Given Given Given 7. 8. 9. 10. 11. t
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