course MTH 151
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21:35:18 Query 3.4.6 write converse, inverse, contrapositive of ' milk contains calcium'
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RESPONSE --> The converse of this statement is: If it contains calcium, then it is milk. (switch order) The inverse of this statement is: If it is not milk, then it does not contain calcium. (keep order, but negate) The contrapositive of this statement is: If it does not contain calcium, then it is not milk. (switch order and negate)
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21:36:26 ** '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 got this one.
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21:43:37 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 contrapositive of this statement would be: If the natural number isn't even, then the square of the natural number isn't even. Ex: Natural number 8 and 9 8 x 8 = 64 9 x 9 = 81 With this example, the origional statement given is true because the square of the number 8 is even therefore the natural number is even. Using the same examples, the contrapositive statement would also be true because the natural number 9 is odd, therefore the square of 81 is odd.
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21:47:40 ** 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 --> I understand here that in the contrapositive statement, it is not indicated which square is not even. I think I missed the wording on this one. I understand the instructors comments you have given.
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21:59:17 Explain how you used examples to determine whether both statements are true or both false.
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RESPONSE --> I used the examples of natural numbers 8 and 9 to determine whether these statements were true or false. The square of 8 is 64, which is an even number as well. The square of 9 is 81. Since the natural number is not even, then it turns out that the square is not even.
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22:00:43 ** 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 --> OK, I did say that both these statements were true, but I don't think I went into as much detail or used as many examples as I should have.
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22:04:53 Explain why either both statements must be true, or both must be false.
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RESPONSE --> I think both must be true because no matter which natural number you choose, the square of that number will be even if the nat number is even or odd if the nat number is odd.
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22:06:59 ** 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, I guess I didn't quite understand what the question was asking. I answered it differently but I do understand that a statement and its contrapositive are identical based on their truth values.
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22:08:41 Query 3.4.24 write 'all whole numbers are integers' in form 'if p then q'.
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RESPONSE --> This statement would read ""If it is a whole number, then it is an integer"" if written in ""if p then q"" form.
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22:09:02 ** 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, I got this one right.
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22:11:26 Query 3.4.30 same for ' principal hires more only if board approves
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RESPONSE --> The books statement says: ""The principal will hire more teachers only if the school board approves"". This statement in ""if p then q"" would read: ""If the principal hires more teachers, then the school board has approved it"", or something along that line.
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22:12:02 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 --> Ok, my statement was worded a little different but I think I got it right.
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22:17:43 Query 3.4.48 true or false: 6 * 2 = 14 iff 9 + 7 neg= 16.
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RESPONSE --> I think this statement is false because for 6 * 2 to be able to equal 14, then the 9 + 7 neg=16 would have to be false. In this case, 9 + 7 is equal to 16, therefore the statement 6 * 2 = 14 is false.
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22:20:12 ** 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 --> Ok, I think I got off on the wrong track when answering this question and comletely missed what it was asking for. I wasn't concentrating on the compound statement, but what it was actually stating.
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22:25:22 Query 3.4.55 contrary or consistent: ' this number is an integer. This number is irrational.'
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RESPONSE --> I think this statement is contrary because for the number to be irrational, it could not be an integer.
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22:25:43 **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 --> I got this one right.
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