WUT FOUNDATION YEAR – winter 2016/2017
Elementary set theory and logic
Recall:
(informally) Predicate – a statement that may be true or false depending on the values of its variables.
p
0
0
1
1
p∨q
0
1
1
1
q
0
1
0
1
p∧q
0
0
0
1
p⇒q
1
1
0
1
p⇔q
1
0
0
1
∀x∈A – for all / for every x in the set A,
∃x∈A – there is / there exists an x in the set A.
3.1. Are the following sets finite of infinite?
(−1)n
: n ∈ N , B = 2k : k ∈ N ,
A=
8
C = [0, 1),
1
D = 2+ : n∈N .
n
3.2. Write as intervals.
A = {x ∈ R : x ≥ 4},
B = {x ∈ R : x < 57},
C = {x ∈ R : x ≤ 0 ∨ x > π},
D = {x ∈ R : x < −1.1 ∧ x ≥ −3.3}, E = x ∈ R : x < π 3 ∨ x > 7 ,
n
√ o
√
3
F = x∈R: x≤ 2 ∧ x> 2 .
3.3. Find the sets.
(a) A =
3
[
(−n, n),
(b) B =
(d) D =
n=1
(−n, n),
(c) C =
1
−1,
,
n
(e) E =
∞
[
[−n, n],
n=1
n=1
n=1
5 \
∞
[
∞ \
n=1
1
−1,
,
n
∞ \
1
(f) F =
−1,
.
n
n=1
3.4. True or false?
(a) a > 0 ⇒ a ≥ 0,
(b) b ≥ 0 ⇒ b > 0,
3.5. True or false?
(a) a < 3 ⇔ a ≤ 3,
(b) |b| > 3 ⇔
√
b2 > 3,
(c) c > 0 ⇒ d2 ≥ 0.
(c) c > 1 ⇔ −5c > −5.
3.6. True or false?
(a) ∀x∈R x2 ≥ 0,
(b) ∀x∈R x2 > 0,
(c) ∃x∈R x2 > 0,
(d) ∃x∈R x2 < 0,
(e) ∃x∈R x2 ≤ 0.
3.7. True or false?
(a)
(c)
(∀x∈R x ≥ 1)
∨
(∀x∈R x < 1) ,
(∃x∈R x ≥ −5)
∧
(∃x∈R x < −5) ,
(b) ∀x∈R
(x ≥ 3 ∨ x < 3) ,
(d) ∃x∈R (x ≥ π ∧ x < π) .
3.8. Write negations of the following predicates.
(a) ∀x∈A x > 0,
(b) ∃y∈B y 5 < 112,
(c) ∀t∈C ∃s∈D t + s ≥ π.