TD Sheet 1: Mathematical Logic
Exercise 7
Let \( x \) and \( y \) be two real numbers. Show that:
\((x \neq 1 \land y \neq 2) \Rightarrow (xy - 2x - y + 2 \neq 0)\)
Exercise 8
Using mathematical induction, prove that if \( x \) is a positive real number, then:
\( \forall n \in \mathbb{N}^*, (1 + x)^n \geq 1 + nx \)
Exercise 9
Prove by induction that for all \( n \in \mathbb{N}^* \),
\[ \sum_{k=1}^n k^2 = 1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n + 1)(2n + 1)}{6}. \]
Last modified: Friday, 12 September 2025, 5:54 PM