Fall 2007
ARE211
Problem Set #04 Answer key
First Graphical Problem Set
(1) Some quick multiple choice questions on necessary and sufficient conditions.
(a) The most important question to ask in determining whether
x
is sufficent for
y
is:
(i) Can you have
y
without having
x
?
(ii) Does
x
have any relevance for
y
?
(iii) Does having
x
ensure that you will have
y
?
(iv) If you know that you have
y
, do you know that you will have
x
?
Justify your answer using the fact that if
x
is sufficient for
y
then the set of objects
satisfying
x
is contained in the set of objects satisfying
y
.
Ans:
The correct answer is
(1(a)iii)
. We know that if
x
is sufficient for
y
then the set
of objects satisfying
x
is contained in the set of objects satisfying
y
. Therefore, having
y
implies having
x
.
(b) The most important question to ask in determining whether
p
is necessary for
q
is:
(i) Can you have p without having q?
(ii) Does having p ensure that you will have q?
(iii) Can p and q both be true at the same time?
(iv) Do you have to have p in order to have q?
Justify your answer using the fact that if
p
is necessary for
q
then the set of objects
satisfying
p
contains the set of objects satisfying
q
.
Ans:
The correct answer is
(1(b)iv)
. We know that if
p
is necessary for
q
then the set of
objects satisfying
p
contains the set of objects satisfying
q
. Therefore, if
q
is satisfied then
p
is satisfied, i.e., you have to have
p
in order to have
q
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
Ans:
(2) Consider the following three class of triangles:
(A) Isoceles
(B) Equilateral
(C) Rightangled
Now answer the following questions
(a) Which class of triangle is sufficient for one of the other classes? Explain your answer.
(b) Which class of triangle is necessary for one of the other classes? Explain your answer.
(c) Is it possible to identify a restriction on one of the angles in the remaining class of
triangles such that if this additional restriction is satisfied, then this class is sufficient
for another class? If yes, identify the condition. If no, explain in terms of set theory.
(d) Is it possible to identify a restriction on one of the angles in the remaining class of
triangles such that if this additional restriction is satisfied, then this class is necessary
for another class? If yes, identify the condition. If no, explain in terms of set theory.
Ans:
(a) (B) is sufficient for (A): if a triangle has three equal sides, then it has two equal sides, and
hence belows to (A);
(b) (A) is necessary for (B): a triangle cannot have three equal sides without having two equal
sides.
(c) Right triangles with one 45
◦
angle must have two sides equal. Hence, the class of Right
angled triangles with a 45
◦
angles is sufficient for the class of isoceles triangles.
(d) Class (C) is already not sufficient for (A). That means that the set consisting of (C) triangles
does not contain the set consisting of (A) triangles. If you add an extra condition to (C),
then you make the set smaller. If (C) doesn’t contain (A) then a subset of (C) certainly
won’t contain (A).
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Simon
 Optimization, Mathematical analysis, Convex function, x*

Click to edit the document details