Homework Week One
Part 1 Economics
1.1 Find examples for each of the following equivalences and draw their truth
tables:
• DeMorgan’s laws
1. ¬(P ∧ Q) ∼ ¬P ∨ ¬Q
2. ¬(P ∨ Q) ∼ ¬P ∧ ¬Q
• Distributive laws
1. P ∧ (Q ∨ R) ∼ (P ∧ Q) ∨ (P ∧ R)
2. P ∨ (Q ∧ R) ∼ (P ∨ Q) ∧ (P ∨ R)
• Absorption laws
1. P ∧ (P ∨ Q) ∼ P
2. P ∨ (P ∧ Q) ∼ P
1.2 Have a look at the following excursion that will be covered on Wednesday. Then prove the last proposition.
Def. E.1 Let X, Y be sets. Then the set
X × Y := {(x, y)|x ∈ X ∧ y ∈ Y }
is called the Cartesian Product of X and Y . We define the Equality of pairs as
follows:
(x, y) = (x0 , y 0 ) :⇔ x = x0 ∧ y = y 0
For the interested reader: We defined the pair (x,y) out of thin air. It would be
much nicer to construct it from the theory of sets, we have developed so far.
What definition based on sets would also fulfill the equality of pairs? Hint:
(x,y) := {x,. . . }
Def. E.2 Let A, B be sets. A subset R ⊆ A×B is called Relation between A and B.
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We can write xRy instead of (x, y) ∈ R.
Rem. E.3 A mapping f : A → B between two sets A and B is defined by three
components:
(a) the domain A
(b) the codomain B
(c) the mapping rule given by f
Def. E.4 Let A, B sets, a mapping f : A → B is a triple (A, B, F ) with:
(i) F ⊆ A × B
(ii) ∀a ∈ A there exists one and only one b ∈ B with (a, b) ∈ F .
For a ∈ A. The unique element b ∈ B with (a, b) ∈ F is denoted by f (a).
Exp. E.5 Define A := {1, 2}, B := {4, 5, 6}.
(1) We define f : A → by F = {(1, 4), (2, 5)}. Then we have f (1) = 4 and
f (2) = 5.
(2) F := {(1, 4), (1, 5)} ⊆ A × B. Now (ii) is not fulfilled.
(3) F := {(1, 4), (2, 4)} ⊆ A × B is a mapping.
(4) Let’s look at f : R → R given by F := {(x, x2 )|x ∈ R}. Then we have:
f (x) = x2
∀x ∈ R.
The set F is called the Graph of the mapping f .
Def. E.6 Suppose A, B sets, f : A → B a mapping, and A0 ⊆ A and B 0 ⊆ B
subsets. Then
(1) f (A0 ) := {f (a)|a ∈ A0 } ⊆ B is the image of A0 under f
(2)f −1 (B 0 ) := {a ∈ A|f (a) ∈ B 0 } ⊆ A0 is the pre-image of B 0 und f
Theorem Suppose f : A → B is a mapping, (Xi )i∈I a family of subsets of A.
Then we have
!
[
f
Xi
=
i∈I
[
f (Xi )
i∈I
Scratch work:
S
S
f
i∈I Xi and
i∈I f (Xi ) are both sets. So we have to show that both sets
are equal. We use what we know about proof on subsets and show ⊆, ⊇ So we
show both:
!
f
[
Xi
⊆
i∈I
[
f (Xi )
i∈I
!
[
f (Xi ) ⊆ f
i∈I
[
i∈I
2
Xi
.
Both of these statements have the form ∀x ∈ C ⇒ x ∈ D
Proof
S
Xi . Then there exists a ∈ i∈I Xi with b = f (a). FurS
thermore there exists i0 ∈ I with a ∈ Xi0 . So b = f (a) ∈ f (Xi0 ) ⊆ i∈I f (Xi ).
S
S
Therefore f
i∈I Xi ⊆
i∈I f (Xi )
⊆ Suppose b ∈ f
S
i∈I
⊇
Suppose b ∈
S
f (Xi ). Then there exists i0 ∈ I with b ∈ f (Xi0 ). So theS
re exists a ∈ Xi0 with b = f (a). Because a ∈ Xi0 ⊆ i∈I f (Xi ) we have
S
S
S
b = f (a) ∈ i∈I f (Xi ). Therefore we have i∈I f (Xi ) ⊆ f
i∈I Xi .
i∈I
Proposition. Suppose f : A → B is a mapping, and (Yi )i∈I a indexed
family of subsets of B.
1. f −1
T
2. f −1
S
i∈I
T
Yi = i∈I f −1 (Yi )
i∈I
S
Yi = i∈I f −1 (Yi )
1.3 Read Chapters 1 and 2 in the Varian as well as Chapter 5A and 5B in
Mas-Colell. If you can’t find a copy of the Mas-Colell, I can give you a copy of
the pages. See if you can grasp all 11 assumptions on the production set. If not
prepare some questions.
Please hand the homework in on Tuesday (01.11.2016). If you won’t attend
the tutorial on Monday, just leave it with the Institutsekretariat (Grimm/Classen).
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