EXERCISES: LOGIC
Exercise 1. In which cases are the following propositions true?
1. (p ⟹ q) ∧ (𝑝̅ ⟹ q).
̅̅̅̅̅̅̅̅̅̅
2. ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(p ∧ (q
∧ r) ⟺ q.
3. ((p ∨ q) ⟹ r) ⟺ (p ⟹ r) ∧ (q ⟹ r).
Exercise 2. Rewrite the following propositions using disjunctions and conjunctions
1. (𝑝̅ ∧ q) ⟹ r.
2. ̅̅̅̅̅̅̅̅̅̅̅
(p ∨ 𝑞̅) ∧ (s ⟹ t).
̅̅̅̅̅̅̅̅̅̅̅
3. (p
∧ q) ∧ (p ∨ q).
Exercise 3.
1. Simplify:
p ∨ (𝑝̅ ∧ q) ∨ (𝑝̅ ∧ 𝑞̅).
2. Give the negation of the following proposition:
(p ∧ q) ∨ (p ∨ q) ∨ (p ∨ 𝑞̅) ⟹ p.
Exercise 4. Give the negation of the following propositions:
1. ∀ x ∈ R, f(x) ≠ 0.
2. ∀ M > 0, ∃ A > 0, ∀ x ≥ A, f(x) > M.
3. ∀ x ∈ R, f(x) > 0 ⟹ x ≤ 0.
4. ∀ ε > 0, ∃ η > 0, ∀ (x,y) ∈ 𝐼 2 , (| x – y | ≤ η ⟹ | f(x) − f(y) | ≤ ε).
Exercise 5. Determine which of the following propositions are true:
1. ∃ x ∈ R, (x + 1 = 0 and x + 2 = 0).
2. (∃ x ∈ R, x + 1 = 0) and (∃ x ∈ R, x + 2 = 0).
3. ∀ x ∈ R, (x + 1 ≠ 0 or x + 2 ≠ 0).
4. ∃ x ∈ 𝑅∗ , ∀ y ∈ 𝑅 ∗ , ∀ z ∈ 𝑅∗ , z – xy = 0.
5. ∀ y ∈ 𝑅∗ , ∃ x ∈ 𝑅 ∗ , ∀ z ∈ 𝑅∗ , z – xy = 0.
6. ∀ y ∈ 𝑅∗ ,∀ z ∈ 𝑅 ∗ , ∃ x ∈ 𝑅∗ , z – xy = 0.
7. ∃ a ∈ R, ∀ ε > 0, |a |< ε.
8. ∀ ε > 0, ∃ a ∈ R, |a| < ε.
Exercise 6. Prove that:
1. ∀ x ∈ R, |𝑥 − 1| ≤ 𝑥 2 − 𝑥 + 1.
2. ∀ x,y ∈ Q, x + y√2 = 1 ⟺ (x = 1 et y = 0).
3. ∀ x ∈ 𝑅∗+ ; ∀ n ∈ N, (1 + 𝑥)𝑛 ≥ 1+ nx.
Exercise 7. Consider the following proposition P defined by:
𝑎2
∃ a ∈ ]− ∞, 0[, ∀ x ∈ [− 1, + ∞[, x > a ⟹ ((𝑥 2 > ) ou (x ≤ 0))
4
1. Give the negation of P.
2. The proposition is it true? Justify.
Exercise 8. Consider the following proposition:
If the integer (𝑛2 −1) is not divisible by 8, then the integer 𝑛 is even.
1. Grive the contrapositive of this proposition.
2. Prove the contrapositive and conclude.
Exercise 9. Let N ≥ 1 be an integer, 𝑀 > 0 a real number, and 𝑎1 , 𝑎2 , … , 𝑎𝑁 real
numbers. We consider the following proposition P:
𝑀
If 𝑎1 + 𝑎2 + ⋯ + 𝑎𝑁 ≥ M then, there exists i ∈ {1, …, N} such that 𝑎𝑖 ≥ .
𝑁
1. Give the contrapositive of P.
2. Prove P.
Exercise 10. Prove that:
1. √2 is not a rational number.
2. If n ∈ 𝑁 ∗ then 𝑛2 + 1 is not the square of a natural number.
𝑛(𝑛+1)(2𝑛+1)
3. ∀ n ∈ 𝑁 ∗ 1 + 22 + 32 + …+ 𝑛2 =
6
4. ∀ n ∈ 𝑁, 2𝑛 > n.
Exercise 11.
1. Let 𝑛 > 0. Prove that if 𝑛 is the square of an integer, then 2𝑛 is not the square of
an integer.
2. Let 𝑛 be an integer. Give and prove the contrapositive of the following
proposition :
If 𝑛2 is odd, then 𝑛 is odd.
3. Prove that :
∀ n ∈ 𝑁 ∗ , 2𝑛−1 ≤ 𝑛! ≤ 𝑛𝑛 .
Exercise 12. Prove that if n is the sum of two squares, then the remainder when n is
divided by 4 is always different from 3.
