Describe or link to a simple method of solving set theory questions

Question

I have a series of set theory questions to answer and the method I have is to draw out venn diagrams one by one and work them out visually. I have been unsuccessful in my search for 3 set arithmetic questions. Is there a simpler general method or a link to a simpler general method of solving them? Example below:

1. $A \cap C \cap B^\prime \cup A \cap B \cap C^\prime \subseteq (A \cup B \cup C)^\prime$

2. $(A \cup B)^\prime \cup C \cap B^\prime \subseteq C \cup A^\prime \cup B^\prime$

Please describe, or link to, a general method able to quickly decide if expressions such as these are true.

Answer 1

First, you need to be much more careful with your parentheses, and add them when needed to avoid ambiguities.

For example, your Venn diagrams should tell you that $A \cap (B \cup C)$ is not the same thing as $(A \cap B) \cup C$

Indeed, if we look at left hand side of your second expression ... is it $((A \cup B)' \cup C) \cap B'$, or is that $(A \cup B)' \cup (C \cap B')$?

Anyway, once you have used parentheses to disambiguate, you can use some algebraic methods, which you can find online (look for 'Boolean algebra' or 'algebra of sets')

To give an example, if we interpret the abov expression as the latter, we ca do the following:

$(A \cup B)' \cup (C \cap B') =$ (DeMorgan)

$(A' \cap B') \cup (C \cap B') =$ (Distribution)

$(A' \cup C) \cap B' = $ (Commutation)

$(C \cup A') \cap B'$

And since in general $A \cap B \subseteq A \cup B$, we can now say that:

$(C \cup A') \cap B' \subseteq C \cup A' \cup B'$