#$&* course mth 151 I wrestled with Query 2.5.24 for a LONG time before I finally GOT it...WOW. What satisfaction when my brain finally gets it! HAHAI do not understand the last question at all.
.............................................
Given Solution: `a** There are 10 numbers in the set: 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qQuery 2.5.18 n({x | x is an even integer } YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: {2,4,6,8....}
.............................................
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 am still confused on the alpha null concept
.............................................
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. ** STUDENT QUESTION I don’t understand what happened to the other 3 choices for pairing. I got (Foxx, Powers) (Foxx, Charles) (Foxx, Peron) (Myers, Powers) (Myers, Charles) (Myers, Peron) (Madonna, Powers) (Madonna, Charles) (Madonna, Peron) INSTRUCTOR RESPONSE What you listed were ordered pairs, one from the first set and one from the second. In fact you listed the 9 pairs of the 'product set'' A X B, an idea you will encounter later in this chapter. However an ordered pair of elements, one from the first set and one from the second (for example your listing (Madonna, Peron)), is not a one-to-one correspondence. In a 1-1 correspondence every element in the first set must be paired with an element in the second. [ {Foxx, Powers},{Myers, Charles},{Madonna, Perron}] is a one-to-one correspondence between the sets. It tells you who each member of the first set is paired with in the second. [{Foxx, Powers},{Myers,Peron},{Madonna, Charles}] is a different one-to-one correspondence. [{Foxx, Charles},{Myers, Powers},{Madonna, Peron}] is another. [ {Foxx, Charles},{Myers,Peron},{Madonna,Powers}], [{Foxx, Peron},{Myers, Powers},{Madonna,Charles}], and [{Foxx, Peron},{Myers, Charles},{Madonna, Powers}] are three more one-to-one correspondences. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Thought I had this one. Still lost on this one.
.............................................
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 forgot to give the formula for remaining numbers into infinity...but after reading your solution I do understand this ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q2.5.42 show two vert lines, diff lengths have same # of points YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I do not know what to do on this confidence rating #$&*:0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q2.5.42 show two vert lines, diff lengths have same # of points YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I do not know what to do on this confidence rating #$&*:0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** This is a pretty tough question. One way of describing the correspondence (you will probably need to do the construction to understand): Sketch a straight line from the top of the blue line at the right to the top of the blue line at the left, extending this line until it meets the dotted line. Call this meeting point P. Then for any point on the shorter blue line we can draw a straight line from P to that point and extend it to a point of the longer blue line, and in our 1-1 correspondence we match the point on the shorter line with the point on the longer. From any point on the longer blue line we can draw a straight line to P; the point on the longer line will be associated with the point we meet on the shorter. We match these two points. If the two points on the long line are different, the straight lines will be different so the points on the shorter line will be different. Thus each point on the longer line is matched with just one point of the shorter line. We can in fact do this for any point of either line. So any point of either line can be matched with any point of the other, and if the points are different on one line they are different on the other. We therefore have defined a one-to-one correspondence. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&! ********************************************* Question: `q2.5.42 show two vert lines, diff lengths have same # of points YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I do not know what to do on this confidence rating #$&*:0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!#*&!