2023/2024 Tutorial=1 Department: Common Core in Mathematics
Analysis 1 Real Numbers and Computer Science Batna 2-University.
Exercise 1
Given x, y, z ∈ R, Prove the following inequalities:
1. |x + y| ≤ |x| + |y|, 1 2
4. (x + y 2 ) ≥ xy
2
2. ||x| − |y|| ≤ |x − y|
p
3. x2 + y 2 ≤ |x| + |y| 5. xy + xz + yz ≤ x2 + y 2 + z 2
Exercise 2
Show that:
√
1. 3 is irrational
√
2. for all (a, b) ∈ Q × Q∗ , the numbers a + b 3 are irrational.
ln 3
3. is irrational.
ln 2
Exercise 3
Justify whether the following assertions are true or false :
a. The sum, the product of two rational numbers, the inverse of a non-zero rational number is a
rational number.
b. The sum or product of two irrational numbers is an irrational.
c. The sum of a rational number and an irrational number is an irrational.
d. The product of a rational number and an irrational number is an irrational.
Exercise 4
Let x, y ∈ R, show that:
1. f (x) = E(x) is an increasing function.
2. E(x) + E(y) ≤ E(x + y) ≤ E(x) + E(y) + 1
E(nx)
3. ∀n ∈ N∗ , E( ) = E(x)
n
Exercise 5
For each of the following sets, describe the set of all upper bounds for the set :
1. the set of odd integers;
1
2. 1 − : n ∈ N ;
n
3. {r ∈ Q : r3 < 8};
4. {sin x : x ∈ R}
Exercise 6
For each of the sets in (1),(2),(3) of the preceding exercise, find the least upper bound of the set, if
it exists.
Exercise 7
Let A, B be two non-empty bounded parts of R. Note −A = {−x, x ∈ A}. Show that:
1. sup(−A) = − inf(A)
2. inf(−A) = − sup(A)
3. If A ⊂ B, then:
sup(A) ≤ sup(B)
inf(B) ≤ inf(A)
4. sup(A ∪ B) = max(sup(A), sup(B))
5. inf(A ∪ B) = min(inf(A), inf(B))
Exercise 8
Determine ( if they exist ) sup, inf, max, min of the following sets :
1. A = [1, 2] ∩ Q 4. D = {x ∈ R : x2 ≤ 3}
2. B = [1, 2[∩Q
5. E = {x ∈ R : |x| > 1}
1
3. C = vn = , n∈N 6. F = {x ∈ R : |x2 − 1| > 1}
n+1
Exercise 9
Let a, b ∈ Q such that a < b, Show that:
∃c∈Q: a<c<b
