2. Define a relation R on pairs of real numbers as follows: (a,
b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a
partial order? Why or why not? If R is a partial order, draw a
diagram of some of its elements.
3. Define a relation R on integers as follows: mRn iff m + n is
even. Is R a partial order? Why or why not? If R is...
The Cantor set, C, is the set of real numbers r for which
Tn(r) ϵ [0,1] for all n, where T is the tent
transformation. If we set C0= [0,1], then we can
recursively define a sequence of sets Ci, each of which
is a union of 2i intervals of length 3-i as
follows: Ci+1 is obtained from Ci by removing
the (open) middle third from each interval in Ci. We
then can define the Cantor set by
C= i=0...
Suppose we define a relation ~ on the set of nonzero real
numbers R* = R\{0} by for all a , b E R*, a ~ b if and only if
ab>0. Prove that ~ is an equivalence relation. Find the
equivalence class [8]. How many distinct equivalence classes are
there?
1. Consider the function f: R→R, where R represents the set of
all real numbers and for every x ϵ R, f(x) = x3. Which of the
following statements is true?
a. f is onto but not one-to-one.
b. f is one-to-one but not onto.
c. f is neither one-to-one nor onto.
d. f is one-to-one and onto.
2. Consider the function g: Z→ {0, 1, 2, 3, 4, 5}, where Z
represents the set of all integers and for...
a. Consider d on R, the real line, to be d(x,y) =
|x2 – y2|. Show that d is NOT a metric on R.
b.Consider d on R, the real line, to be d(x,y) =
|x3 – y3|. Show that d is a metric on R.
2. Let d on R be d(x,y) = |x-y|. The “usual”
distance. Show the interval (-2,7) is an open set.
Note: you must show that any point z
in the interval has...
Using field axioms, prove the following theorems:
(i) If x and y are non-zero real numbers, then xy does not equal
0
(ii) Let x and y be real numbers. Prove the following
statements
1. (-1)x = -x
2. (-x)y = -(xy)=x(-y)
3. (-x)(-y) = xy
(iii) Let a and b be real numbers, and x and y be non-zero real
numbers. Then a/x + b/y = (ay +bx)/(xy)
Consider the following economy with: Real Money demand 〖
(M/P)〗^d = – 12 R + 0.38 Y Real Money supply (M^s/P)= 4510 Derive
the LM curve Derive the LM curve when the money supply increases by
680. Derive the LM curve when money supply decreases by 12% Compare
the LM curves from a, b and c by graphing them using any graphing
tool (excel preferably). Comment on the differences. Find the value
of money demanded when income Y = 15,000...
5. Determine whether or not the following functions from real
numbers to real numbers are bijections. If they are bijections,
then find the inverse. If they are not bijections, then explain why
not.
(a) f(x) = [2x]
(b) f(x) = −7x
(c) f(x) = 7x 3 – 5
(d) f(x) = x 2 − 5