course MTH 151 I think I may have forgotten to put my ID in the last one so I'm resubmitting it. 005. `Query 5
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Given Solution: `a** There are 10 numbers in the set: 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qQuery 2.5.18 n({x | x is an even integer } YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: All even integers fit into this set; the number is infinite. confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: `a** {x | x is an even integer } indicates the set of ALL possible values of the variable x which are even integers. Anything that satisfies the description is in the set. This is therefore the set of even integers, which is infinite. Since this set can be put into 1-1 correspondence with the counting numbers its cardinality is aleph-null. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I forgot to say it was “aleph-null” ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `qQuery 2.5.24 how many diff corresp between {Foxx, Myers, Madonna} and {Powers, Charles, Peron}? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Fox <-> Charles , Madonna <-> Peron , Myers <-> Powers Fox <-> Peron , Madonna <-> Charles , Myers <-> Powers Fox <-> Peron, Madonna <-> Powers , Myers <-> Charles Fox <-> Powers , Madonna <-> Peron , Myers <-> Charles Fox <-> Powers , Madonna <-> Charles, Myers <-> Peron Fox <-> Charles, Madonna <-> Powers, Myers <-> Peron There are 6 correspondence confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** Listing them in order, according to the order of listing in the set. We have: [ {Foxx, Powers},{Myers, Charles},{Madonna, Perron}] , [{Foxx, Powers},{Myers,Peron},{Madonna, Charles}], [{Foxx, Charles},{Myers, Powers},{Madonna, Peron}] [ {Foxx, Charles},{Myers,Peron},{Madonna,Powers}], [{Foxx, Peron},{Myers, Powers},{Madonna,Charles}], [{Foxx, Peron},{Myers, Charles},{Madonna, Powers}] for a total of six. Reasoning it out, there are three choices for the character paired with Foxx, which leaves two for the character to pair with Myers, leaving only one choice for the character to pair with Madonna. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I only spelled Fox with one “x” ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q2.5.36 1-1 corresp between counting #'s and {-17, -22, -27, ...} YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The number in the first set decreased by 5 each correspondence. If n = 1 then if we have -12 +5 *n = -17. This works with any of the other number as well Ex. If n = 2 then -12 + 5*n = -22 confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 2
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Given Solution: `a**You have to describe the 1-1 correspondence, including the rule for the nth number. A complete description might be 1 <-> -17, 2 <-> -22, 3 <-> -27, ..., n <-> -12 + 5 * n. You have to give a rule for the description. n <-> -12 - 5 * n is the rule. Note that we jump by -5 each time, hence the -5n. To get -17 when n=1, we need to start with -12. THE REASONING PROCESS TO GET THE FORMULA: The numbers in the first set decrease by 5 each time so you need -5n. The n=1 number must be -17. -5 * 1 = -5. You need to subtract 12 from -5 to get -17. So the formula is -5 n - 12. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I am not sure I answer this correctly but I understand the correspondence rule that I put but -12 - -5 doesn’t work… ------------------------------------------------ Self-critique Rating: 2 ********************************************* Question: `q2.5.42 show two vert lines, diff lengths have same # of points YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I drew the diagram from the book then made a line that connected them to a point on horizontal line. Anything in the angle that attaches to one of the vertical lines should have an exact corresponding point on the other. I don’t know how to explain it better…
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Given Solution: `a** This is a pretty tough question. One way of describing the correspondence (you will probably need to do the construction to understand): Sketch a straight line from the top of the blue line at the right to the top of the blue line at the left, extending this line until it meets the dotted line. Call this meeting point P. Then for any point on the shorter blue line we can draw a straight line from P to that point and extend it to a point of the longer blue line, and in our 1-1 correspondence we match the point on the shorter line with the point on the longer. From any point on the longer blue line we can draw a straight line to P; the point on the longer line will be associated with the point we meet on the shorter. We match these two points. If the two points on the long line are different, the straight lines will be different so the points on the shorter line will be different. Thus each point on the longer line is matched with just one point of the shorter line. We can in fact do this for any point of either line. So any point of either line can be matched with any point of the other, and if the points are different on one line they are different on the other. We therefore have defined a one-to-one correspondence. **