course Mth 151 x{{ع롴˭}Oassignment #015
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08:53:44 Query 3.4.6 write converse, inverse, contrapositive of ' milk contains calcium'
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RESPONSE --> Well first I put it into p -> q form, and then I found the following from that converse: if it contains calcium then it's milk inverse: if it's not milk then it doesn't contain calcium contrapositive: if it doesn't contain calcium then it's not milk confidence assessment: 3
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08:54:04 ** '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 self critique assessment: 3
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08:55:55 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 --> if the natural number isn't even , then the square of a natural number isn't even. Those are true. confidence assessment: 3
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08:56:12 ** 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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08:58:28 Explain how you used examples to determine whether both statements are true or both false.
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RESPONSE --> Well the first statement said that if the square of a natural number is even then the natural number is even. 144 is the square of 12. 16 is the square of 4. I could be wrong on that. The contrapositive says that if the natural number is even then its square isn't even like 5^2=25. So the contrapositive is true. confidence assessment: 3
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08:58:49 ** 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 --> OK self critique assessment: 3
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08:59:31 Explain why either both statements must be true, or both must be false.
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RESPONSE --> If one is true, the other has to be true. Same goes with the false side of things. confidence assessment: 2
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08:59:48 ** 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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09:00:13 Query 3.4.24 write 'all whole numbers are integers' in form 'if p then q'.
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RESPONSE --> If it's a whole number, it's an integer. confidence assessment: 3
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09:00:21 ** 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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09:00:39 Query 3.4.30 same for ' principal hires more only if board approves
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RESPONSE --> If the board approves, then the principal hires more confidence assessment: 3
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09:01:16 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 self critique assessment: 3
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09:03:10 Query 3.4.48 true or false: 6 * 2 = 14 iff 9 + 7 neg= 16.
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RESPONSE --> Both statments are false, but the compound statement is true because there are two false statements. confidence assessment: 2
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09:03:30 ** 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 self critique assessment: 3
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09:04:27 Query 3.4.55 contrary or consistent: ' this number is an integer. This number is irrational.'
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RESPONSE --> All integers are rational. So the statement is contrary. confidence assessment: 3
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09:04:34 **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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