course Mth 151 ?????W??????assignment #015015. `query 15
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19:45:07 Query 3.4.6 write converse, inverse, contrapositive of ' milk contains calcium'
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RESPONSE --> if it contains calcium, then it is milk If it is not milk then it does not contain calcium If it doesn't contain calcium, then it is not milk confidence assessment: 3
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19:45:25 ** '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 --> got it! self critique assessment: 3
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19:47:16 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 --> 18) if the natural number is not even, then the square of a nutural number is not even it is true confidence assessment: 3
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19:48:21 ** 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 --> okay, I wasn't sure where you were going with that. But I got it self critique assessment: 3
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19:49:17 Explain how you used examples to determine whether both statements are true or both false.
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RESPONSE --> i just calculated that if it is not an even number then the square cannot be positive confidence assessment: 2
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19:49:25 ** 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 --> self critique assessment:
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19:50:19 Explain why either both statements must be true, or both must be false.
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RESPONSE --> because they will be contrary if one statement is false the other cannot be true. They have to be one or the other confidence assessment: 3
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19:50:43 ** 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 --> that is what i thought self critique assessment: 3
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19:51:04 Query 3.4.24 write 'all whole numbers are integers' in form 'if p then q'.
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RESPONSE --> if it is a whole number, then it is an integer confidence assessment: 3
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19:53:37 ** 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 --> that's what I put. I think I am ready to teach your class..........I am hardly telling the truth. I have a lot to learn before I can even pass this class self critique assessment: 3
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19:54:16 Query 3.4.30 same for ' principal hires more only if board approves
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RESPONSE --> If the school board approves, then the principal will hire more teachers confidence assessment: 3
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19:54:31 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 --> I got it self critique assessment: 3
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19:55:47 Query 3.4.48 true or false: 6 * 2 = 14 iff 9 + 7 neg= 16.
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RESPONSE --> true, they are both false confidence assessment: 3
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19:56:10 ** 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 --> good job zach self critique assessment: 3
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19:57:15 Query 3.4.55 contrary or consistent: ' this number is an integer. This number is irrational.'
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RESPONSE --> contrary, it is saying that one is this and the other is that. confidence assessment: 3
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19:57:27 **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 --> self critique assessment:
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