Sets: suggested exercises

Exercise 1. Tell whether the following sets are bounded from below and/or from above, specifying
infimum, supremum and, minimum and maximum, if they exist.

1
 
a) A = 1 + : n ∈ N \ {0} .
n
n o
b) B = 2 + ncos (nπ) : n ∈ N .
n o
c) C = e1/n : n ∈ N \ {0} .

n+1
 
d) D = : n ∈ N \ {2} .
n−2
1 π
   
e) E = cos n : n ∈ N \ {0} .
n 2
f ) F = {[cos (nπ)]n : n ∈ N}.

g) G = 2 − 2−n : n ∈ N .
1
 
h) H = cos (nπ) − : n ∈ N \ {0} .
n
1
 
i) I = : n ∈ N \ {0, 1} .
log n
j) J = {|en − 2|en : n ∈ N}.

Exercise 2. Tell whether the following sets are bounded from below and/or from above, specifying
infimum, supremum and, minimum and maximum, if they exist.
n p o
a) A = x ∈ R : x2 − 2x ≤ 1 .
n o
b) B = x ∈ R : x2 ≥ 2x .
n o
c) C = e1/x : x ∈ R \ {0} .

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2 Sets: suggested exercises

n o
d) D = x ∈ R : |x| ≥ x2 .
n o
e) E = x ∈ R : 2ex − 1 ≥ e2x .
np o
f) F = x2 − 2x − 1 : x ∈ (−∞, 0] ∪ [2, +∞) .
n o
g) G = 2x − x2 : x ∈ R .

x+1
 
h) H = : x ∈ R \ {2} .
x−2
n o
i) I = |x| − x2 : x ∈ R .
n o
j) J = e2x − 2 ex + 1 : x ∈ R .

Exercise 3. Let A, B ⊆ R two bounded intervals, such that A ∩ B 6= ∅. Prove that

sup(A ∩ B) = min{sup A, sup B}, inf(A ∩ B) = max{inf A, inf B}.

Exercise 4. Determine A \ B and B \ A in the following cases:

1
  n o
1) A = x ∈ R : x > , B = x ∈ R : x2 − 3x ≤ 0
x
2) A = {x ∈ R : |x − 1| ≥ x} , B = {x ∈ R : 2x − 1 < 0}
π
 
3) A = sin x : 0 < x < , B = {2x − 1 : x ≥ 0}
2

Exercise 5. Draw A × B in the following cases:

1) A = [1, 2], B = [−1, 3]

2) A = [1, 2], B = {−1, 3}

3) A = {1, 2}, B = {−1, 3}

4) A = {1, 2}, B = [−1, 3]

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Sets: suggested exercises 3

SOLUTIONS
Exercise 1.

a) A is bounded from above and from below; inf A = 1, sup A = max A = 2, min A does not
exist.

b) B is bounded from below and it is not bounded from above; inf B = min B = 2.

c) C is bounded from above and from below; inf C = 1, sup C = max C = e, min C does not
exist.

d) D is bounded from above and from below; inf D = min D = −2, sup D = max D = 4.

e) E is bounded from above and from below; inf E = min E = − 21 , sup E = max E = 41 .

f ) F = {−1, 1} is bounded from above and from below; inf F = min F = −1, sup F = max F =
1.

g) G is bounded from above and from below; inf G = min G = 1, sup G = 2, max G does not
exist.

h) H is bounded from above and from below; inf H = min H = −2, sup H = 1, max H does not
exist.

1
i) I is bounded from above and from below; inf I = 0, sup I = max I = log 2 , min I does not
exist.

j) J is bounded from below and it is not bounded from above; inf J = min J = 1.

Exercise 2.
h √ i h √ i √
a) A = 1 − 2, 0 ∪ 2, 1 + 2 is bounded from above and from below; inf A = min A = 1− 2,
√
sup A = max A = 1 + 2.

b) B = (−∞, 0] ∪ [2, +∞) is bounded nor from below neither from above.

c) C = (0, 1) ∪ (1, +∞) is bounded from below, it is not bounded from above, inf C = 0, min C
does not exist.

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4 Sets: suggested exercises

d) D = [−1, 1] is bounded from above and from below; inf D = min D = −1, sup D = max D =
1.

e) E = {0} is bounded from above and from below; inf E = min E = sup E = max E = 0.

f ) F = [−1, +∞) is bounded from below, it is not bounded from above; inf F = min F = −1.

g) G = (−∞, 1] is bounded from above, it is not bounded from above; sup G = max G = 1.

h) H = (−∞, 1) ∪ (1, +∞) is bounded nor from below neither from above.
 i
i) I = −∞, 14 is bounded from above, it is not bounded from below; sup I = max I = 41 .

j) J = [0, +∞) is bounded from below, it is not bounded from above; inf J = min J = 0.

Exercise 3. Since A and B are bounded intervals such that A ∩ B 6= ∅, A ∩ B either contains a
single point or is an interval.
In the first case the point is necessarily the infimum of one of the two intervals and the
supremum of the other one, that is x0 = sup A = inf B or x0 = inf A = sup B. Then x0 =
sup(A ∩ B) = min{sup A, sup B} and x0 = inf(A ∩ B) = max{inf A, inf B}.
If A ∩ B is a nonempty interval, there are two possible cases:

1) A ⊆ B (or B ⊆ A). Then A∩B = A (or A∩B = B), therefore sup(A∩B) = min{sup A, sup B}
and inf(A ∩ B) = max{inf A, inf B};

2) A 6⊆ B and B 6⊆ A. In this case A ∩ B is an interval whose end-points are m, M with m < M .

Obviously m = inf(A ∩ B) and M = sup(A ∩ B). We denote by ma = inf A, Ma = sup A,
mb = inf B, Mb = sup B.

Therefore ma ≤ mb = m ≤ Ma = M ≤ MB or mb ≤ ma = m ≤ Mb = M ≤ MA .

Thus
m = max{inf A, inf B}, M = min{sup A, sup B}.

Exercise 4.

1) A = (−1, 0) ∪ (1, +∞) and B = [0, 3]. Hence A \ B = (−1, 0) ∪ (3, +∞) and B \ A = [0, 1].
 i   n o
2) A = −∞, 12 and B = −∞, 12 . Hence A \ B = 1
2 and B \ A = ∅.

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Sets: suggested exercises 5

3) A = (0, 1) and B = [0, +∞). Hence A \ B = ∅ and B \ A = {0} ∪ [1, +∞).

Exercise 5.

1) A × B is the rectangle with edges (1, −1), (2, −1), (2, 3) and (1, 3).

O 1 2 x

−1

2) A × B is the union of two segments, the first one with end-points (1, −1) and (2, −1), the
other one with end-points (1, 3) and (2, 3).

O 1 2 x

−1

3) A × B is the set containing the four points of coordinates (1, −1), (2, −1), (2, 3) and (1, 3).

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6 Sets: suggested exercises

b b
3

O 1 2 x

b b
−1

4) A × B is the union of two segments, the first one with end-points (1, −1) and (1, 3), the other
one with end-points (2, −1) and (2, 3).

O 1 2 x

−1

c 2017 Politecnico di Torino