course mth 151 thank you for allowing me to share time with my grandmother before she passed. I'm a bit behind but I'm catching up. P遣assignment #015
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10:16:11 Query 3.4.6 write converse, inverse, contrapositive of ' milk contains calcium'
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RESPONSE --> If it is milk then is contains calcium if it is not milk then it does not contain calcium if it does not contain calcium then it is not milk
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10:16:33 ** '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
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10:41:19 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 is not even then the square of a natural number is not even
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10:41:22 ** 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
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10:42:35 Explain how you used examples to determine whether both statements are true or both false.
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RESPONSE -->
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10:42:38 ** 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
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10:43:33 Explain why either both statements must be true, or both must be false.
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RESPONSE --> because it is either true of false, if one is true the other must be true if one if false so is the other
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10:43:37 ** 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
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10:44:36 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 a integer
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10:44:40 ** 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
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10:46:06 Query 3.4.30 same for ' principal hires more only if board approves
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RESPONSE --> if the principal hires more then the board approves
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10:46:09 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
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10:47:34 Query 3.4.48 true or false: 6 * 2 = 14 iff 9 + 7 neg= 16.
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RESPONSE --> false
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10:47:39 ** 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
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10:48:02 Query 3.4.55 contrary or consistent: ' this number is an integer. This number is irrational.'
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RESPONSE --> contrary
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10:48:05 **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