16 pts) #4. TRUE/FALSE. Determine the truth value of each sentence (no explanation required). ________(a) A statement...
16 pts)
#4. TRUE/FALSE. Determine the truth value of each sentence
(no explanation required).
________(a) A statement is a sentence that is
true.
________(b) In logic, p q refers to the
"inclusive or, " true when either p or q or both are
true.
________(c) The phrase "not p and not q" means
"not both p and q."
________(d) The conditional statement p q is
true if p is false.
________(e) The negation of p q is
p ~q.
#5. Write the logical negation of each of the following
statements in a clearly worded unambiguous English sentence.
(a) Carol visited a beach resort or Carol visited a mountain
resort.
(b) If Steve is not a full-time student, then Steve has a
full-time job.
Quantifiers (#7-8, 12 pts)
#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.
(a) Some athletes earn at least one million dollars per
year.
(b) No drivers are toddlers.
(c) All distances are positive.
Homework Answers
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16 pts) #4. TRUE/FALSE. Determine the truth value of each sentence (no explanation required). ________(a) A statement...
#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....
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Quantifiers, Counterexamples, Disproof (#9, 15 pts) #9. For each statement, state whether the statement is true...
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Answer the following questions in the space provided below. 1. For each proposition below, first determine...
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2. Indicate whether each statement is true or false. If False, rewrite the sentence so that...
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Note whether the sentence is true of false on the line following each sentence. If a...
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Decide whether each statement is always true or always false. Give a short (1–3 sentence) explanation why with specific reference to theorems/definitions/facts/etc. from the book. 0 [1 12. The matrix 0 -2 2 -87 is an REF of the matrix 0 14 -5 8 07 -1 1 1 1 1 LO 0 14. -1 0 0 13. The vectors [3, -1, 10) and (3, -2, 5] form a basis for span([1,0,5), (3, -1, 10), [3, -2,5]).
true and false propositions with quantifiers. Answer the following questions in the space provided below. 1. For each proposition below, first determine its truth value, then negate the proposition and simplify (using De Morgan's laws) to eliminate all – symbols. All variables are from the domain of integers. (a) 3.0, x2 <. (b) Vr, ((x2 = 0) + (0 = 0)). (c) 3. Vy (2 > 0) (y >0 <y)). 2. Consider the predicates defined below. Take the domain to...
Quantifiers, Counterexamples, Disproof (#9, 15 pts) #9. For each statement, state whether the statement is true or false. If false, explain; provide a counterexample as appropriate or a careful explanation. (If true, no explanation expected) (d)x, y in R, if Ixl < lyl, then x<xy. (e) 3 m in N such that V n in N, msn (f)n in N, 3x in R such that n <x. 3x in R such that v n in N, Vn<x. (g)
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Write true or false for each of the following statements. Provide justification for each answer—if true, give a brief explanation. If false, either provide a counterexample or contrast the statement with a similar true statement, explaining why the two cases differ. (5 points) The functions ePX and eq* are linearly independent when p + q.
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Decide whether each statement is always true or always false. Give a short (1–3 sentence) explanation why with specific reference to theorems/definitions/facts/etc. from the book. 0 [1 12. The matrix 0 -2 2 -87 is an REF of the matrix 0 14 -5 8 07 -1 1 1 1 1 LO 0 14. -1 0 0 13. The vectors [3, -1, 10) and (3, -2, 5] form a basis for span([1,0,5), (3, -1, 10), [3, -2,5]).
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