Fundamental Concepts

13 Dec 2012

I am trying something I have seen on many blogs around the internet: I will work through all the exercises in Munkres’ textbook, Topology. I have never tried anything like this before and it will definitely test my ability to stay motivated as I tend to get bored easily. So, without further ado, I present the exercises of Topology, Chapter 1, “Set Theory and Logic”, Section 1, “Fundamental Concepts”.

1. Check the distributive laws for $\cup$ and $\cap$ and DeMorgan’s laws.

This amounts to showing the following are true:

• $A \cap ( B \cup C ) = (A \cap B) \cup (A \cap C)$
• $A \cup ( B \cap C ) = (A \cup B) \cap (A \cup C)$
• $A - (B \cup C) = (A - B) \cap (A - C)$
• $A - (B \cap C) = (A - B) \cup (A - C)$

$A \cap (B \cup C) = \{ x \mid x \in A \mbox{ and } x \in (B \cup C) \}$