Download Econ. 700 Tauchen/Petranka Summer 2008 Homework #1 For

Econ. 700
Tauchen/Petranka
Summer 2008
Homework #1
For questions 1 - 7a, explain each step including use of the basic algebraic properties of the
reals (e.g., distributive property of multiplication). In the remaining problems, explain every step
except (i) the commutative, associative, and distributive properties of multiplication, addition, and
subtraction of the reals, (ii) the existence of the additive inverse for any real number, (iii) the
multiplicative inverse for any non-zero real number, and (iv) the closure properties of the natural
numbers, integers, rationals, and irrationals.
1. Let w, x, y, and z be real numbers with w < x and y < z. Show that w + y < x + z. (This
property of inequalities is often used.)
2. Show that the sum of a rational number and an irrational number is irrational.
√
3. The text includes a proof that the square root of two, denoted 2, exists. Use a proof by contradiction to show that the square root of two is irrational. [Hint: For a proof by contradiction,
we begin by assuming that the square√root of two is rational rather than irrational. Thus, there
exist integers p and q 6= 0 such that 2 = p/q.1 If the integer p were zero, then the square of
the right hand side of the equality would be zero whereas the square of the left hand side is 2.
Thus p 6= 0.
a. What properties of the reals and equalities are needed in order to show that p2 = 2q 2 ?
b. Is the number p2 prime or composite? Does p2 have an even or odd number of prime
factors? Does q 2 have an even or odd number of prime factors? Does the number 2q 2 have
an even or odd number of prime factors? Is the number of prime factors for p2 and 2q 2
consistent with the two numbers being equal?
c. Which of the above steps would not hold if you tried to use the method to show that the
square root of 25 was irrational? Explain.]
4. For any positive real number x show that there exists a natural number M such that 1/M < x.
5. Let x and y be positive real numbers with y − x > 1. Show that there is an integer between x
and y.
6. Let z be a real number. Show that there exists an irrational number larger than z.
7. Let X be a set and {Sn } = {S1 , S2 , S3 , . . .} be subsets of X. By definition, the sequence of sets
is nested if Sn+1 ⊆ Sn where n is any positive integer.
a. Let X be the set of reals and Sn be the open interval (1 − 1/n, 1 + 1/n). Show that the
only element of
∞
\
Sn
n=1
is the number 1.
1
A term of the form x/y for real numbers x and y 6= 0 means x multiplied times the multiplicative inverse of y
(which we know exists). The term x/y could be written equivalently as x · 1/y or x · y −1 where 1/y and y −1 are
common symbols for the multiplicative inverse of y.
b. Construct a nested sequence of nonempty sets with an empty intersection. [Hint: The
sequence above was constructed about the point 1, and the intersection of any such sequence
of sets includes the point about which the sets are constructed. Instead of beginning with
the set (0,2) and “taking away” points from both sides of the interval for sets S2 , S3 , . . .,
try taking away points only from one side.]
c. Construct a nested sequence of unbounded sets with an empty intersection.
8. Let a be a positive number. Prove by induction that (1 + a)n ≥ 1 + na for any natural number
n.
9. Let a and b be positive integers with b 6= 0. Show that there exist integers r and q with 0 ≤ r < b
such that a = bq + r.
[Hint: Define the set
S = {n|n is a natural number and nb ≤ a}.
a. Prove that S is nonempty.
b. Prove that S is bounded above.
c. Explain why S necessarily has a maximum element, which we will call q.
d. Define r = a − bq. Show that 0 ≤ r < b.]
10. Show that
n
X
k 2 = n(n + 1)(2n + 1)/6.
k=1
11. Let f : X ⊆ R → Y ⊆ R, or in other words let f be a function from a domain which is a subset
of the reals to a range Y which is a subset of the reals . For any set S ⊆ X, define the set f (S)
as {y ∈ Y |f (x) = y for some x ∈ S}.
Let f : R → R be the function f (x) = x2 for x ∈ R. Describe the set f ([1, 2]) where [1,2] is the
closed interval from [1,2]. Describe the set f ([a, b]) where a and b are real numbers with a < b.
12. Let g : X ⊆ R → Y ⊆ R. Let S1 and S2 be subsets of X. Show that f (S1 ) ∪ f (S2 ) = f (S1 ∪ S2 ).
Does f (S1 ) ∩ f (S2 ) = f (S1 ∩ S2 )? Explain.
13. The set S ⊆ R has the property that for any element x ∈ S there exists an element y ∈ S such
that y > x. Is the set S necessarily unbounded?
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