function f whose domain is S and whose target is T. Let a be an
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Concept explainers
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
Transcribed Image Text:PROBLEM 2
Ten kids line up for recess. The names of the kids are:
{Alex, Bobby, Cathy, Dave, Emy, Frank, George, Homa, Ian, Jim}.
Let S be the set of all possible ways to line up the kids. For example, one order
might be:
(Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby)
The names are listed in order from left to right, so Frank is at the front of the
line and Bobby is at the end of the line.
Let T be the set of all possible ways to line up the kids in which George is ahead
of Dave in the line. Note that George does not have to be immediately ahead of
Dave. 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 r be an
element of S, so x is one possible way to order the kids. If George is ahead of Dave
in the ordering r, then f(x) = r. If Dave is ahead of George in a, then f(r) is the
ordering that is the same as r, except that Dave and George have swapped places.
(a) What is the output of f on the following input?
(Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby)
(b) What is the output of f on the following input?
(Emy, Ian, Dave, Homa, Jim, Alex, Bobby, Frank, George, Cathy)
(c) Is the function f a k-to-1 correspondence for some positive integer k? If
so, for what value of k? Justify your answer.
(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|.
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