ECON 381 SC 2012 ASSIGNMENT 1 (CORRECTED) Answer all the ...

ECON 381 SC 2012 ASSIGNMENT 1
(CORRECTED)
JOHN HILLAS
UNIVERSITY OF AUCKLAND
Answer all the following problems. Each problem is worth the same number of
marks (20). For questions with multiple parts each part is worth the same number
of marks. Some parts are however significantly harder and longer than others.
Problem 1. Use a truth table to find whether the following statements are equivalent.
Explain precisely why the truth table implies the result that you state.
((p ⇒ r)∨(q ⇒ p)) and ((p∧q) ⇒ r).
Problem 2. Let X = {a,b,c,d,e}. In the following questions use this set X.
(i) Write out explicitly what the set X×X is. Explain what we mean by a binary
relation on X and define this explicitly and formally.
(ii) Give an example of a binary relation on X that is complete and one that is
not complete.
(iii) Give an example of a binary relation that is transitive but not negatively
transitive.
(iv) Give an example of a binary relation that is negatively transitive but not
transitive.
(v) What is an equivalence? Give an example of an equivalence on X.
Problem 3. Let X = {x,y,z} and Y = {a,b,c,d,e}
(i) Give an example of a function f : X → Y that is one-to-one, that is, that is
an injection.
(ii) Is the function that you have given onto, that is, a surjection?
(iii) What is the range of the function you have given?
(iv) If we let the range of f be denoted f(X) consider the function ˜f : X → f(X)?
Is this function one-to-one or onto (or both)?
(v) Does ˜f have an inverse? If so find that inverse, if not explain why the inverse
does not exist.
Problem 4. Consider the following “metric” (“metric” is in quotes since, until you
do part (i) of the problem, you don’t know that this is actually a metric) on the
set of all n−tuples of real numbers:
{
0 if x = y,
d D (x,y) =
1 otherwise.
This metric is called the discrete metric.
(i) Prove that for any set X the discrete metric d D is a metric.
(ii) Let X be any set and consider the metric space (X,d D ). What are the open
subsets of X with this metric? You should prove your answer not just state
what the set is.
Date: Second Semester, 2011.
1

2 JOHN HILLAS UNIVERSITY OF AUCKLAND
Problem 5. Let (X,d) be a metric space.
Another approach to putting structure of a set X is to directly specify the open
sets. Such a space is called a topological space. If (X,τ) is a topological space then
X is a set and τ is a collection of subsets of X such that
1. X ∈ τ
2. ∅ ∈ τ
3. If A and B are in τ then A∩B is in τ. That is, the intersection of two elements
of τ is in τ. (This also implies that the intersection of any finite number of
elements of τ is in τ.)
4. If A n is in τ for every n ∈ N for some set N that might be finite or infinite then
∪ n∈N A n is in τ. That is, the union of any number of elements of τ is in τ.
Now let X = {a,b,c,d,e}. For the questions below use this particular set X.
(i) Show that for any metric d the set of open subsets of X are the same. Find
what this collection of open sets is.
(ii) Show that if we let τ = {X,∅} that this collection of “open sets” satisfies the
four properties given above.
(iii) Show that this topology cannot come from any metric. (Hint: you have
essentially already show this in your answer to part (i).)
(iv) Give another set τ that satisfies the four properties given above.
(v) Characterise the set of all possible topologies τ on X that satisfy the four
properties.