course Mth 151 V|—ðCò©vkæÌáÊcŸý˜©`„½ýœ…i·]éÛ£Äassignment #015
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11:27:35 Query 3.4.6 write converse, inverse, contrapositive of ' milk contains calcium'
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RESPONSE --> p: milk q: contains calcium Converse: q -> p If it contains calcium, then it is milk. Inverse: ~p -> ~q If it isn't milk then it doesn't contains calcium Contrapostive: ~q -> ~p If it doesn't contain calcuim, then is isn't milk. confidence assessment: 2
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11:29:28 ** '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 --> The converse is just switching the orginal statement; the inverse you keep the order and neg. the statement and contrapostive you swith the order of the inverse statement. self critique assessment: 2
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11:37:19 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 --> p: square of nature even number q: even natural number Contrapostive: ~q -> ~p If a natural number is not even, then the square of the natural number is not even. 2^2 = 4; 4^2=16; 6^2=36 True confidence assessment: 2
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11:41:55 ** 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 --> The contrapostive is the neg of the original switched. self critique assessment: 2
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11:43:17 Explain how you used examples to determine whether both statements are true or both false.
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RESPONSE --> I did the sqaure roots of some numbers and determined it was true because they came out even. confidence assessment: 2
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11:44:01 ** 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 worked out squares of some even #s. self critique assessment: 2
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11:46:08 Explain why either both statements must be true, or both must be false.
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RESPONSE --> Because the truth tables show that the orginal and the contrapostive of the orig statement have the same truth values. confidence assessment: 3
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11:47:35 ** 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 --> I remembered that the truth tables states that they have the same truth values but I worked the problem out orginially instead of remembering the truth table. self critique assessment: 2
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11:49:15 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: 2
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11:49:58 ** 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 --> I did not indentify p and q before answering. self critique assessment: 2
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11:54:51 Query 3.4.30 same for ' principal hires more only if board approves
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RESPONSE --> p: principal q: board If the principal hires more teachers, the borad approved it. confidence assessment: 2
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11:56:40 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 --> The error comment says that the principal will hire the teachers before the board approves. The board has to approve first according to the original statement. self critique assessment: 2
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12:02:03 Query 3.4.48 true or false: 6 * 2 = 14 iff 9 + 7 neg= 16.
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RESPONSE --> It is true because both statements are false. confidence assessment: 2
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12:02:41 ** 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 --> The biconditional table for FF is T. self critique assessment: 2
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12:09:59 Query 3.4.55 contrary or consistent: ' this number is an integer. This number is irrational.'
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RESPONSE --> Contrary because a integer is not an irrational #. confidence assessment: 2
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12:10:37 **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 --> Both statements weren't true so it is contrary. self critique assessment: 2
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