course Mth 151 跆ׁɹŻ蒾assignment #006
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22:35:37 Query 1.1.4 first 3 children male; conclusion next male. Inductive or deductive?
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RESPONSE --> Inductive - the statement goes from specific to general.
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22:35:49 ** The argument is inductive, because it attempts to argue from a pattern. **
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RESPONSE --> ok
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22:36:27 Query 1.1.8 all men mortal, Socrates a man, therefore Socrates mortal.
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RESPONSE --> Deductive - the statement starts out general and goes into a specific statement.
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22:36:53 ** this argument is deductive--the conclusions follow inescapably from the premises. 'all men' is general; 'Socrates' is specific. This goes general to specific and is therefore deductive. COMMON ERROR: because it is based on a fact, or concrete evidence. Fact isn't the key; the key is logical inevitability. The argument could be 'all men are idiots, Socrates is an man, therefore Socrates is an idiot'. The argument is every bit as logical as before. The only test for correctness of an argument is that the conclusions follow from the premises. It's irrelevant to the logic whether the premises are in fact true. **
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RESPONSE --> ok
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22:38:23 Query 1.1.20 1 / 3, 3 / 5, 5/7, ... Probable next element.
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RESPONSE --> From the book, the answer is 11/13. Using your example the answer is 7/9.
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22:39:01 **The numbers 1, 3, 5, 7 are odd numbers. We note that the numerators consist of the odd numbers, each in its turn. The denominator for any given fraction is the next odd number after the numerator. Since the last member listed is 5/7, with numerator 5, the next member will have numerator 7; its denominator will be the next odd number 9, and the fraction will be 7/9. There are other ways of seeing the pattern. We could see that we use every odd number in its turn, and that the numerator of one member is the denominator of the preceding member. Alternatively we might simply note that the numerator and denominator of the next member are always 2 greater than the numerator and denominator of the present member. **
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RESPONSE --> ok
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22:41:13 Query 1.1.23 1, 8, 27, 64, ... Probable next element.
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RESPONSE --> I am going to guess that the answer will be an odd number because the other numbers are alternating odd, even, odd, even. The answer is 125 because that is what the book says, but I couldn't figure this one out.
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22:43:49 ** This is the sequence of cubes. 1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125. The next element is 6^3 = 216. Successive differences also work: 1 8 27 64 125 .. 216 7 19 37 61 .. 91 12 18 24 .. 30 6 6 .. 6 **
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RESPONSE --> I see how the answer was obtained, but I never would have thought of using cubes. Does that mean to the third power?
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22:45:34 Query 1.1.36 11 * 11 = 121, 111 * 111 = 12321 1111 * 1111 = 1234321; next equation, verify.
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RESPONSE --> 11111 x 11111 = 123,454,321 The pattern is building on the ones. Two ones (11), three ones (111), four ones (1111) and so on. The answers look pretty neat, too.
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22:45:41 ** We easily verify that 11111*11111=123,454,321 **
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RESPONSE --> ok
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22:49:11 Do you think this sequence would continue in this manner forever? Why or why not?
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RESPONSE --> No, the product will continue to build with 123 at the beginning and 321 at the end and then the middle numbers will begin to change and then that will change the end numbers. I think 123 will remain at the beginning.
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22:49:41 ** You could think forward to the next few products: What happens after you get 12345678987654321? Is there any reason to expect that the sequence could continue in the same manner? The middle three digits in this example are 8, 9 and 8. The logical next step would have 9, 10, 9, but now you would have 9109 in the middle and the symmetry of the number would be destroyed. There is every reason to expect that the pattern would also be destroyed. **
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RESPONSE --> ok
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22:51:09 Query 1.1.46 1 + 2 + 3 + ... + 2000 by Gauss' method
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RESPONSE --> 1 + 2000 = 2001 2001 x 1000 (number of pairs) = 2,001,000
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22:51:23 ** Pair up the first and last, second and second to last, etc.. You'll thus pair up 1 and 2000, 2 and 1999, 3 and 1998, etc.. Each pair of numbers totals 2001. Since there are 2000 numbers there are 1000 pairs. So the sum is 2001 * 1000 = 2,001,000 **
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RESPONSE --> ok
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22:54:49 Query 1.1.57 142857 * 1, 2, 3, 4, 5, 6. What happens with 7? Give your solution to the problem as stated in the text.
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RESPONSE --> 142,857 x 7 = 999,999 The pattern that was developing from x 1 through x 6 was a flip flop pattern. x 1 and x 6 mirrored each other (142,857 and 857,142), x 2 and x 5 mirrored each other (285,714 and 714,285) and x 3 and x 4 mirrored each other (428,571 and 571,428).
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22:55:08 ** Multiplying we get 142857*1=142857 142857*2= 285714 142857*3= 428571 142857*4=571428 142857*5= 714285 142857*6=857142. Each of these results contains the same set of digits {1, 2, 4, 5, 7, 8} as the number 1428785. The digits just occur in different order in each product. We might expect that this pattern continues if we multiply by 7, but 142875*7=999999, which breaks the pattern. **
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RESPONSE --> ok
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22:57:23 What does this problem show you about the nature of inductive reasoning?
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RESPONSE --> Many times, inductive reasoning starts out ""logically,"" but then it changes. Basically, don't put faith in inductive reasoning. You could be correct and you more than likely will assume incorrectly.
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22:57:33 ** Inductive reasoning would have led us to expect that the pattern continues for multiplication by 7. Inductive reasoning is often correct it is not reliable. Apparent patterns can be broken. **
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RESPONSE --> ok
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