course Mth 151 I know I am still running behind on assignments. I plan to get everything up to date this week. Thanks! ???????????_vassignment #014
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10:25:38 `q001. There are 8 questions in this set. If each of the propositions p and q can be either true or false, what combinations of truth values are possible for the two propositions (e.g., one possibility is that p is false and q is true; list the other possibilities)?
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RESPONSE --> p q F T T T T F F F confidence assessment: 0
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10:25:58 It is possible that p is true and q is true. Another possibility is that p is true and q is false. A third possibility is that p is false and q is true. A fourth possibility is that p is false and q is false. These possibilities can be listed as TT, TF, FT and FF, where it is understood that the first truth value is for p and the second for q.
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RESPONSE --> ok self critique assessment: 3
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10:29:07 `q002. For each of the for possibilities TT, TF, FT and FF, what is the truth value of the compound statement p ^ q ?
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RESPONSE --> p^q T F F F confidence assessment: 0
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10:29:17 p ^ q means 'p and q', which is only true if both p and q are true. In the case TT, p is true and q is true so p ^ q is true. In the case TF, p is true and q is false so p ^ q is false. In the case FT, p is false and q is true so p ^ q is false. In the case FF, p is false and q is false so p ^ q is false.
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RESPONSE --> ok self critique assessment: 3
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10:30:18 `q003. Write the results of the preceding problem in the form of a truth table.
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RESPONSE --> p q p^q T T T T F F F T F F F F confidence assessment: 0
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10:30:28 The truth table must have headings for p, q and p ^ q. It must include a line for each of the possible combinations of truth values for p and q. The table is as follows: p q p ^ q T T T T F F F T F F F F.
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RESPONSE --> ok self critique assessment: 3
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10:31:58 `q004. For each of the possible combinations TT, TF, FT, FF, what is the truth value of the proposition p ^ ~q?
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RESPONSE --> p^~q F T F T confidence assessment: 0
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10:33:44 For TT we have p true, q true so ~q is false and p ^ ~q is false. For TF we have p true, q false so ~q is true and p ^ ~q is true. For FT we have p false, q true so ~q is false and p ^ ~q is false. For FF we have p false, q false so ~q is true and p ^ ~q is false.
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RESPONSE --> OK self critique assessment: 3
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10:35:27 `q005. Give the results of the preceding question in the form of a truth table.
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RESPONSE --> p q ~q p^~q T T F F T F T T F T F F F F T F confidence assessment: 0
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10:35:38 The truth table will have to have headings for p, q, ~q and p ^ ~q. We therefore have the following: p q ~q p^~q T T F F T F T T F T F F F F T F
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RESPONSE --> ok self critique assessment: 3
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10:37:50 `q006. Give the truth table for the proposition p U q, where U stands for disjunction.
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RESPONSE --> p q pUq T T T T F T F T T F F F confidence assessment: 0
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10:37:58 p U q means 'p or q' and is true whenever at least one of the statements p, q is true. Therefore p U q is true in the cases TT, TF, FT, all of which have at least one 'true', and false in the case FF. The truth table therefore reads p q p U q T T T T F T F T T F F F
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RESPONSE --> ok self critique assessment: 3
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10:42:37 `q007. Reason out the truth values of the proposition ~(pU~q).
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RESPONSE --> ~(pU~q) is true confidence assessment: 0
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10:44:19 In the case TT p is true and q is true, so ~q is false. Thus p U ~q is true, since p is true. So ~(p U ~q) is false. In the case TF p is true and q is false, so ~q is true. Thus p U ~q is true, since p is true (as is q). So ~(p U ~q) is false. In the case FT p is false and q is true, so ~q is false. Thus p U ~q is false, since neither p nor ~q is true. So ~(p U ~q) is true. In the case FF p is false and q is false, so ~q is true. Thus p U ~q is true, since ~q is true. So ~(p U ~q) is false.
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RESPONSE --> OK. I did this one wrong. I see what I was supposed to do. self critique assessment: 3
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10:47:08 `q008. Construct a truth table for the proposition of the preceding question.
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RESPONSE --> p q ~q ~(pU~q) T T F F T F T F F T F F F F T F confidence assessment: 0
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10:49:00 We need headings for p, q, ~q, p U ~q and ~(p U ~q). Our truth table therefore read as follows: p q ~q pU~q ~(pU~q) T T F T F T F T T F F T F F T F F T T F
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RESPONSE --> I left out the heading for pU~q. I got alittle messed up on this one, but I think I see what I needed to do and where I messed up. self critique assessment: 3
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course Mth 151 ???}R????Q????????assignment #015015. `query 15
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11:03:08 Query 3.4.6 write converse, inverse, contrapositive of ' milk contains calcium'
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RESPONSE --> calcium contains milk milk does not contain calcium calcium does not contain milk confidence assessment: o
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11:04:26 ** 'Milk contains calcium' can be put into p -> q form as 'if it's milk then it contains calcium'. The converse of p -> q is q -> p, which would be 'if it contains calcium then it's milk' The inverse of p -> q is ~p -> ~q, which would be 'if it's not milk then it doesn't contain calcium'. The contrapositive of p -> q is ~q -> ~p, which would be 'if it doesn't contain calcium then it's not milk'. Note how the original statement and the contrapositive say the same thing, and how the inverse and the converse say the same thing. NOTE ON ANOTHER STATEMENT: If the statement is 'if it ain't broke don't fix it: Converse: If you don't fix it, then it ain't broke Inverse: If it's broke, then fix it. Contrapositive: If you fix it, then it's broke. **
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RESPONSE --> OK. I went out in left field on this one, but I'm back now. I think I understand what I need to do. self critique assessment: 3
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11:10:59 Query 3.4.18 state the contrapositive of 'if the square of the natural number is even, then the natural number is even.' Using examples decide whether both are truth or false.
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RESPONSE --> The natural number is not even if the square of the natural number is not even. If the conditional statement is true then the contrapositve statement always has the same value. confidence assessment: 0
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11:12:05 ** The statement is of the form p -> q with p = 'square of nat number is even' and q = 'nat number is even'. The contrapositive of p -> q is ~q -> ~p, which in this case would read 'if a natural number isn't even then its square isn't even'. STUDENT RESPONSE WITH SOMEWHAT PICKY BUT IMPORTANT INSTRUCTOR CORRECTION: if the natural number isn't even , then the square of a natural numbewr isn't even Good. More precisely: if the natural number isn't even , then the square of THAT natural number isn't even. To say that the square of a natural number isn't even doesn't necessarily refer to the given uneven natural number. COMMON ERROR WITH INSTRUCTOR COMMENT: The natural number is not even, if the square of a natural number is not even. ex.-3^2=9,5^2=25 This statement is true. ** You have stated the inverse ~p -> ~q. It doesn't matter that the 'if' is in the second half of your sentence, the 'if' in your statement still goes with ~p when it should go with ~q. COMMON ERROR WITH INSTRUCTOR COMMENT: If the natural number is not even, then the square of the natural number is not even. This statement does not involve square roots. It addresses only squares. And 26 isn't the square of a natural number. **
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RESPONSE --> ok self critique assessment: 3
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11:13:40 Explain how you used examples to determine whether both statements are true or both false.
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RESPONSE --> I guess I did it wrong because I didn't use examples. I went on the statement that if the conditional statement is true (or false) then the contrapositive will always have the same truth value. I guess I did it incorrectly?? confidence assessment: 0
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11:14:49 ** The first statement said that if the square of a natural number is even then the natural number is even. For example, 36 is the square of 6, 144 is the square of 12, 256 is the square of 16. These examples make us tend to believe that the statement is true. The contrapositive says that if the natural number is even then its square isn't even. For example, the square of the odd number 7 is 49, which is not an even number. The square of the odd number 13 is 169, which is not an even number. This and similar examples will convince us that this statement is true. **
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RESPONSE --> I see I should have actually been checking for and showing my results for square even #'s. self critique assessment: 3
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11:16:26 Explain why either both statements must be true, or both must be false.
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RESPONSE --> I believe I already did that (when I wasn't supposed too) about how the conditional statement, whether true or false, will have the same contrapositive truth value. confidence assessment: 0
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11:16:37 ** The reason is that the truth tables for the statement and its contrapositive are identical, so if one is true the other is true and if one is false the other must be false. **
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RESPONSE --> ok self critique assessment: 3
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11:19:37 Query 3.4.24 write 'all whole numbers are integers' in form 'if p then q'.
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RESPONSE --> If a number is a whole number then it is an integer. confidence assessment: 0
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11:19:50 ** p could be 'it's a whole number' and q would then be 'it's an integer'. The statement would be 'if it's a whole number then it's an integer'. **
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RESPONSE --> ok self critique assessment: 3
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11:21:46 Query 3.4.30 same for ' principal hires more only if board approves
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RESPONSE --> If the school board approves, the principal will hire more teachers. confidence assessment: o
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11:22:46 COMMON ERROR WITH INSTRUCTOR COMMENT: If the principal will hire more teachers, then the school board would approve. INSTRUCTOR COMMENT: p only if q is the same as if p then q; should be 'if the principle hires, the school board approved' **
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RESPONSE --> So, was my response wrong? self critique assessment: 3
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11:24:37 Query 3.4.48 true or false: 6 * 2 = 14 iff 9 + 7 neg= 16.
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RESPONSE --> 9+7 does equal 16 so the statement is false. confidence assessment: 0
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11:25:58 ** Both statments are false, but the compound statement is true. The compound statement 'p if and only if q' is equivalent to 'if p then q, AND if q then p'. This compound statement is true because p and q are both false, so 'if p then q' and 'if q then p' are both of form F -> F and therefore true **
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RESPONSE --> I did not include whether the compound statement was T of F, but I understand that if both statements are F then the compound is T. self critique assessment: 3
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11:34:27 Query 3.4.55 contrary or consistent: ' this number is an integer. This number is irrational.'
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RESPONSE --> The statement is contrary. confidence assessment: 0
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11:34:45 **Any integer n can be expressed in the form p / q as n / 1. So all integers are rational. Irrational numbers are defined as those numbers which are not rational. So the statements are indeed contrary-it is impossible for a number to be both an integer and irrational. **
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RESPONSE --> ok self critique assessment: 3
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