Lecture 14 - Exercises
Part A - Proving
Non-Conditional Statements
- Prove the following proposition: Proposition:
Suppose \(a \in \mathbb{Z}\). Then
\(14 \mid a\) if and only if \(7 \mid a\) and \(2 \mid a\).
- Prove the following proposition: Proposition: There
exists a positive real number x for which \(x^2 < \sqrt{x}\).
- Prove the following proposition:
\[
\forall n \in \mathbb{N}, \sum_{i=0}^n 2^i = 2^{i+1} - 1
\]
Part B - Mechanics of
Disproof
- What must you show in order to disprove a claim of the form \(\forall x \in S, P(x)\)?
- What must you show in order to disprove a claim of the form \(P(x) \Rightarrow Q(x)\)?
- What must you show in order to disprove a claim of the form \(\exists x \in S, P(x)\)?
- How would you disprove \(P\) with
proof by contradiction?
Part C - Proof/Disproof
Examples
- Prove or disprove the following: Proposition: If
\(x, y \in R\) and \(|x + y| = |x − y|\), then \(y = 0\).
- Prove or disprove the following: Proposition: Every
odd integer is the sum of three odd integers.
- Prove or disprove the following: Proposition: There
exist two prime numbers \(p\) and \(q\) for which \(p
− q = 97\).