Ten kids line up for recess. The names of the kids are: {Abe, Ben,...

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Ten kids line up for recess. The names of the kids are:
{Abe, Ben, Cam, Don, Eli, Fran, Gene, Hal, Ike, Jan}.
Let S be the set of all possible ways to line up the kids. For example, one ordering might be:
(Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben)
The names are listed in order from left to right, so Fran is at the front of the line and Ben is at the end of the line.
Let T be the set of all possible ways to line up the kids in which Gene is ahead of Don in the line. Note that Gene does not have to be immediately ahead of Don. For example, the ordering shown above is an element in T.
Now define a function f whose domain is S and whose target is T. Let x be an element of S, so x is one possible way to order the kids. If Gene is ahead of Don in the ordering x, then f(x) = x. If Don is ahead of Gene in x, then f(x) is the ordering that is the same as x, except that Don and Gene have swapped places.
(a)
What is the output of f on the following input?
(Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben)
(b)
What is the output of f on the following input?
(Eli, Ike, Don, Hal, Jan, Abe, Ben, Fran, Gene, Cam)
(c)
Is the function f a kto1 correspondence for some positive integer k? If so, for what value of k?
(d)
There are 3628800 ways to line up the 10 kids with no restrictions on who comes before whom. That is, S = 3628800. Use this fact and the answer to the previous question to determine T.
(e)
Let Q be the set of orderings in which Gene comes before Don and Jan comes before Abe (again, not necessarily immediately before). Define a kto1 correspondence from S to Q. Use the value of k to determine Q.
Answer & Explanation
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(a) (Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben). [f(x)=x]
(b) (Eli, Ike, Gene, Hal, Jan, Abe, Ben, Fran, Don, Cam).
(c) Yes. k= 2.
(d) T = 1814400 .
(e) Given x in T , define g to be a function from T to Q such that if Jan comes before Abe in x then f(x) = x. If Abe comes before Jan then swap the position of Jan and Abe keeping everyone else's position as it was in x. Finally given y in S, define a function h from S to Q such that h(y) = g(f(y)) where f is the same function as before.
Q = 907200.
Stepbystep explanation
(a) Notice that Gene is ahead of Don in the line up (Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben).
Hence f((Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben)) = (Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben). [f(x)=x]
(b) Notice that Don is ahead of Gene in the line up (Eli, Ike, Don, Hal, Jan, Abe, Ben, Fran, Gene, Cam). So we swap the positions of Gene and Don to get, f((Eli, Ike, Don, Hal, Jan, Abe, Ben, Fran, Gene, Cam)) = (Eli, Ike, Gene, Hal, Jan, Abe, Ben, Fran, Don, Cam).
(c) Yes. Consider a lineup x= (x1,.......,x10) in T. Notice that for each x in T, f(x) = x. Let 1<=i<j<=10 be such that xi= Gene, xj=Don. Now consider a lineup y= (y1,.....,y10) in S.
Then Notice that if there exists any
l∈{0,⋯,10}∖{i,j}
such that xl != yl Then f(y) != x.
So f(x)= y implies yl=xl for all l∈{0,⋯,10}∖{i,j}
Now notice that if yi = Gene, then we have y = x. and f(y) = f(x)=x.
If yi = Don. then since i<j, hence when f is applied position of Don and Gene gets swapped. Hence, if f(y) = (x'1,......,x'10) then x'i = Gene = xi and x'j = Don = xj. Hence x'= x.
So, we have k= 2.
(d) we have for each x in T there exists exactly 2 y in S such that f(y) =X. Hence S= 2T. So T = 3628800/2 = 1814400 .
(e) Given x in T , define g to be a function from T to Q such that if Jan comes before Abe in x then f(x) = x. If Abe comes before Jan then swap the position of Jan and Abe keeping everyone else's position as it was in x. Finally given y in S, define a function h from S to Q such that h(y) = g(f(y)). Doing similar analysis as in (c) we show that both g and f are 2 to 1 functions. Hence, we conclude that h is a 4 to 1 function i.e. k =4.
Further S = 2T = 4Q. Hence Q = 3628800/4 = 907200.