Analysis 1: Exercises 2
1.
Let
A, B
be sets from a universe. Prove the following properties of the complement.
(a) (
A
c
)
c
=
A
.
(b) (
A
∪
B
)
c
=
A
c
∩
B
c
.
(c) (
A
∩
B
)
c
=
A
c
∪
B
c
.
2.
Let
A, B
and
C
be sets. Prove the following equalities
(a)
A
4
B
= (
A
∪
B
)

(
A
∩
B
).
(b) (
A
4
B
=
∅
)
⇔
(
A
=
B
).
3.
What do the following statements mean? Do you think they true or false?
(a) (
∀
x
∈
N
)(
∃
y
∈
N
)(
x < y
).
(b) (
∃
y
∈
N
)(
∀
x
∈
N
)(
x < y
).
(c) (
∃
x
∈
N
)(
∀
y
∈
N
)(
x < y
).
(d) (
∀
y
∈
N
)(
∃
x
∈
N
)(
x < y
).
(e) (
∃
x
∈
N
)(
∃
y
∈
N
)(
x < y
).
(f) (
∀
x
∈
N
)(
∀
y
∈
N
)(
x < y
).
4.
Analyse the logical form of the following statements.
(a) Jane saw a bear and Roger saw one too.
(b) Jane saw a bear and Roger saw it too.
You could use, for example,
P
(
x, y
) to stand for the proposition ”
x
saw
y
”.
5.
Let
P
(
x
) be the statement that says that a real number
x
has some property
P
.
Rewrite the following propositions using connectives and quantifiers. Construct
negations to each of them.
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 Winter '10
 Spyros
 Set Theory, Multiplication, Sets

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