Case 3.ATay-Sachs Disease
A 6-month-old infant girl presented to the emergency room with her first seizure. Her parents
reported that she had not had any fever, nor had she had any vomiting, diarrhea,
positive integers n. 28. Let b be a fixed
integer and j a fixed positive integer. Show
that if P (b), P (b + 1), . . . , P (b + j ) are
true and [P (b) P (b + 1) P (k)] P
(k + 1)is true for every inte
nonnegative integer. 6. Prove that 1 1! +
2 2!+ n n! = (n + 1)! 1 whenever n
is a positive integer. 7. Prove that 3 + 3 5
+ 3 52+ 3 5n=3(5n+1 1)/4
whenever n is a nonnegative integer. 8.
Prove that 2
property can be used to show that there is
a unique greatest common divisor of two
positive integers. Let a and b be positive
integers, and let S be the set of positive
integers of the form as + bt, w
INDUCTIVE STEP: We show that the
conditional statement [P (1) P (2)
P (k)] P (k + 1) is true for all positive
integers k. Note that when we use strong
induction to prove that P (n) is true for all
po
into (n2 + n + 2)/2 regions if no two of
these lines are parallel and no three pass
through a common point. 63. Let a1,
a2,.,an be positive real numbers. The
arithmetic mean of these numbers is
define
recursively defined set unless it is in the
initial collection specified in the basis step
or can be generated using the recursive
step one or more times. Later we will see
how we can use a technique
k, then P (k + 1) is true. That is, for the
inductive hypothesis we assume that P (j )
is true for j = 1, 2,.,k. The validity of both
mathematical induction and strong
induction follow from the wellor
string containing no symbols).
RECURSIVE STEP: If w and x , then
wx . GABRIEL LAM (17951870)
Gabriel Lam entered the cole
Polytechnique in 1813, graduating in
1817. He continued his education at the
c
of the ladder. However, there is no obvious
way to complete this inductive step
because we do not know from the given
information that we can reach the (k +
1)st rung from the kth rung. After all, we
why these steps show that this inequality
is true whenever n is an integer greater
than 1. 20. Prove that 3n< n! if n is an
integer greater than 6. 21. Prove that 2n >
n2 if n is an integer greater th
attempt to prove P (n) for all integers
nwith n 3 using strong induction, the
inductive step does not go through. b)
Show that we can prove that P (n) is true
for all integers n with n 3 by proving by
must successively make to break the bar
into n separate squares. Use strong
induction to prove your answer. 11.
Consider this variation of the game of
Nim. The game begins with n matches.
Two players
whenever n is an integer greater than or
equal to 3. In Exercises 47 and 48 we
consider the problem of placing towers
along a straight road, so that every
building on the road receives cellular
servic
If k + 1 is prime, we immediately see that
P (k + 1) is true. Otherwise, k + 1 is
composite and can be written as the
product of two positive integers a and b
with 2 a b 3. Consider the first three
el
this stable assignment. Use strong
induction to show that the deferred
acceptance algorithm produces a stable
assignment that is optimal for suitors. 25.
Suppose that P (n) is a propositional
function
Case 7.A Edema
James is a 72-year old man with chronic renal failure that recently suffered
from a heart attack. He underwent angioplasty and was receiving IV fluids.
During the night a nurse on the 1
Case 2.AHyperbilirubinemia
Bilirubin is a yellow pigment formed when hemoglobin is broken down or red
blood cells have reached the end of their lifespan.
There are two forms of bilirubin, direct and i
Case 6.AStress Response
You are in the woods taking a hike when you encounter a bear with its cub.
Your heart starts pounding and you get ready to run away. You have initiated
the fight or flight resp
used to prove the stronger inequality 1 2
3 4 2n 1 2n < 1 3n + 1 for all
integers greater than 1, which, together
with a verification for the case where n =
1, establishes the weaker inequality we
or
integers n, because (1) tells us P (1) is
true, completing the basis step and (2)
tells us that P (1) P (2) P (k)
implies P (k + 1), completing the inductive
step. Example 1 illustrates how strong
ind
of a golf course with an infinite number of
holes and that if this golfer plays one hole,
then the golfer goes on to play the next
hole. Prove that this golfer plays every
hole on the course. Use math
and the last two lines must meet in a
common point p2. But in this case, p1 and
p2 do not have to be the same, because
only the second line is common to both
sets of lines. Here is where the inductive
Assume that whenever max(x, y) = k and x
and y are positive integers, then x = y.
Now let max(x, y) = k + 1, where x and y
are positive integers. Then max(x 1, y
1) = k, so by the inductive hypothesi
triangles in the triangulation have two
sides that border the exterior of the
polygon. 18. Use strong induction to
show that when a simple polygon P with
consecutive vertices v1, v2, ., vn is
triangul
b. 5. State what needs to be proved under
the assumption that the inductive
hypothesis is true. That is, write out what
P (k + 1) says. 6. Prove the statement P (k
+ 1) making use the assumption P (k)
standing in the line with heights that
are either increasing or decreasing.
23. Show that whenever 25 girls and
25 boys are seated around a circular
table there is always a person both of
whose neighb