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Introduction to Proofs

Homework (V)

The next three problems are from Chapter 2 of the Textbook You can find the original versions as Problem 2.6., Problem 2.14. and Problem 2.15.

Problem 1

For each of the following statements, determine whether it is true for all sets $X,Y$ (with a proof), false for all sets $X,Y$ (with a proof), or true for some choices of $X$ and $Y$ and false for others (with a counter-example). Here $\mathcal{P}$ stands for the power-set operation.

  • $\mathcal{P}(X \cup Y) = \mathcal{P}(X) \cup \mathcal{P}(Y)$
  • $\mathcal{P}(X \cap Y) = \mathcal{P}(X) \cap \mathcal{P}(Y)$
  • $\mathcal{P}(X \times Y) = \mathcal{P}(X) \times \mathcal{P}(Y)$
  • $\mathcal{P}(X \setminus Y) = \mathcal{P}(X) \setminus \mathcal{P}(Y)$

Problem 2

A subset $U \subseteq \mathbb{R}$ is said to be open if, for all $a \in U$, there exists $\delta > 0$ such that $(a-\delta, a+\delta) \subseteq U$. For each of the following subsets of $\mathbb{R}$, determine (with proof) whether it is open:

  • $\varnothing$
  • $(0,1)$
  • $(0,1]$
  • $\mathbb{Z}$
  • $\mathbb{R} \setminus \mathbb{Z}$

Problem 3

Prove that a subset $U \subseteq \mathbb{R}$ is open if and only if, for all $a \in U$, there exist $u,v \in \mathbb{R}$ such that $u<a<v$ and $(u,v) \subseteq U$.

Problem 4

Prove the following identities of sets. Do you need the Law of Excluded Middle in any of your arguments? Identify where LEM is used in your proofs.

  • $A \setminus \displaystyle \bigcup_{i\in I} X_i = \bigcap_{i\in I} (A \setminus X_i)$
  • $A \setminus \displaystyle \bigcap_{i\in I} X_i = \bigcup_{i\in I} (A \setminus X_i)$