Let $n\ge1\in\Bbb N$, we define the set of binary boolean vectors with $\varphi^n .$
Prove that $\varphi^n$ is a boolean algebra.
So (...)
I know that:
Let $\varphi=\{0,1\}, \mathrm A=(a_1,..,a_n), \mathrm B=(b_1,..,b_n); \mathrm {A,B} \in \varphi^n$
Then, $S=(\varphi^n,+,\cdot,')$ is a boolean algebra if and only if:
- $\mathrm{A+B}=(a_1+b_1,..,a_n+b_n)$
- $\mathrm{A\cdot B}=(a_1\cdot b_1,..,a_n \cdot b_n)$
- $\mathrm{A'}=(a_1',..,a_n')$
But, how can I prove this?
Also I have to prove that $\varphi^n$ is biyective? because $||\mathrm{A}||$ and $||\mathrm{B}||$ must be the same.
THANKS!
The exercise asks you to define 3 operations and two constants $\mathbf 0$ and $\mathbf 1$ in $\varphi^n$, and then prove that they satisfy the axioms of a boolean algebra. You seem to propose the 3 operations, now try to find the 2 constants that will satisfy the following axioms:
The first one is the associativity for the two binary laws. For example, prove that $(A+B)+C = A+(B+C)$ (this isn't difficult, because this follows from associativity of addition modulo 2).
The second one is commutativity, i.e $A+B = B+A$ and $A \cdot B = B\cdot A$, but again this is trivial.
A list of the axioms can be found here (where $\wedge$ means $\cdot$ and $\vee$ means $+$ in your case): http://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Definition Try to prove the rest of the axioms.
Concerning your question, it doesn't mean anything to say that "$\varphi^n$ is bijective", it is not a function but a set of vectors, and the length of $A$ and $B$ are the same since you took them as length $n$ vectors. Maybe you misphrased your question, could you elaborate?