I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding Boolean algebra. To be specific, I'm stuck on the following practice question:
Let $A = \{a, b\}$ and list the four elements of the power set $\mathcal{P}(A)$. We consider the operation $+$ to be $\cup$, $\cdot$ to be $\cap$, and complement to be set complement. Consider $1$ to be $A$ and $0$ to be $∅$.
- Explain why the description above defines a Boolean algebra.
- Find two elements $x$, $y$ in $\mathcal{P}(A)$ such that $xy = 0$, $x \neq 0$ and $y \neq 0$.
Starting with the power set: $$\mathcal{P}(A) = \{∅, \{a\},\{b\},\{a,b\}\}.$$
How would I go about finding the elements of $x$ and $y$ to satisfy part two of the question using algebraic axioms? Also, for explaining how the above defines a Boolean algebra, do you think it would suffice to simply mention how there are two binary operations and a set associated with the Boolean algebra?
Best Answer
Let $x=\{a\}$ and $y=\{b\}$. We have $x\ne\emptyset$ and $y\ne\emptyset$. We also have $xy=x\cap y=\{a\}\cap\{b\}=\emptyset=0$, so that seems to work.
A boolean algebra has variables that can only be true or false (or $1$ or $0$). Normal algebraic operations can continue as usual, unless of course they don't work.