#$&* course Mth 151 1/28 11:15 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: `a** The argument is inductive, because it attempts to argue from a pattern. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `qQuery 1.1.8 all men mortal, Socrates a man, therefore Socrates mortal. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This reasoning is Deductive. It starts with a general, factual statement then specifies. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** 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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 1.1.20 1 / 3, 3 / 5, 5/7, 7/9, ... Probable next element. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The element is 9/11. Each time 2 is added to the previous number. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a**The numbers 1, 3, 5, 7, 9 and 11 are odd numbers. We note that the numerators consist of the odd numbers, each in its turn. The denominator for any given fraction is the next odd number after the numerator. Since the last member listed is 7/9, with numerator 7, the next member will have numerator 9; its denominator will be the next odd number 11, and the fraction will be 9/11. There are other ways of seeing the pattern. We could see that we use every odd number in its turn, and that the numerator of one member is the denominator of the preceding member. Alternatively we might simply note that the numerator and denominator of the next member are always 2 greater than the numerator and denominator of the present member. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I got the right answer, but I went about it in a different way than the ones listed. ------------------------------------------------ Self-critique Rating:2 ********************************************* Question: `qQuery 1.1.23 This problem wasn't assigned, but you should be able to make a good attempt: 1, 8, 27, 64, ... What is the probable next element? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I don’t know. confidence rating #$&*: 0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** 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 ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Oh, I see. ------------------------------------------------ Self-critique Rating: o ********************************************* Question: `qQuery 1.1.36 11 * 11 = 121, 111 * 111 = 12321 1111 * 1111 = 1234321; next equation, verify. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 11111 * 11111 = 123454321 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** We easily verify that 11111*11111=123,454,321 ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qDo you think this sequence would continue in this manner forever? Why or why not? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Ummm… Yeah, I think that it would continue on forever. There is no indication that it wouldn’t. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** 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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Oh! Whoops. I didn’t think about it that way. ------------------------------------------------ Self-critique Rating: 0 ********************************************* Question: `qQuery 1.1.46 1 + 2 + 3 + ... + 2000 by Gauss' method YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I can’t find anything about Gauss’ method in my book. I’m gonna guess that the sum would be 4,000. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** Pair up the first and last, second and second to last, etc.. You'll thus pair up 1 and 2000, 2 and 1999, 3 and 1998, etc.. Each pair of numbers totals 2001. Since there are 2000 numbers there are 1000 pairs. So the sum is 2001 * 1000 = 2,001,000 ** Self-critique (if necessary) : Whoa! I’m way off. ------------------------------------------------ Self-critique Rating: 0 ********************************************* Question: `qQuery 1.1.55 (previously 1.1.57) 142857 * 1, 2, 3, 4, 5, 6. What happens with 7? Give your solution to the problem as stated in the text. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 1 * 142857 = 142857 2 * 142857 = 285714 3 * 142857 = 428571 4 * 142857 = 571428 5 * 142857= 714285 6 * 142857 = 857142 Each of these answers are a grouping of the original number 142857 only in different orders. 7 * 142857 = 999999 Oh, neat! It completely changes all of the sudden. confidence rating #$&*: 2. I give myself this confidence rating because I’m not sure that I correctly figured out the pattern. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** 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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Oh, I wasn’t wrong after all!! ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `qWhat does this problem show you about the nature of inductive reasoning? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It shows that what looks like an obvious pattern and answer might not always be right. It’s best to prove it by deductive reasoning before one reaches a conclusion. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** Inductive reasoning would have led us to expect that the pattern continues for multiplication by 7. Inductive reasoning is often correct it is not reliable. Apparent patterns can be broken. ** " end document Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qWhat does this problem show you about the nature of inductive reasoning? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It shows that what looks like an obvious pattern and answer might not always be right. It’s best to prove it by deductive reasoning before one reaches a conclusion. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** 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. ** " end document Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!