Boolean algebra truths with more than one digit

Question

I understand that when you 0 AND 0 it will result in 0 and that 1 AND 1 will result in 1 etc but what I don't understand is if the question was 01 AND 11. How would I work this out?

Answer 1

Since these operations are defined per pair of boolean values, you should apply them for each boolean pairwise. In your example this will be:
b₀ b₁ ​ AND ​ b₂ b₃ ​ ​ ​ = ​ ​ ​ (b₀ AND b₂) ​ (b₁ AND b₃)

Answer 2

This notation is usually used for bitwise operations. You compute the operation bit by bit – the $i$th bit of the answer is (in this case) the AND of the $i$th bits of the inputs.

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From a more theoretical side of things, this bitwise rule does turn $k$-bit numbers into a Boolean algebra. Boolean algebras are sets together with several operations that satisfy certain axioms (there are several equivalent versions which are easy to find online). Stone's representation theorem states that every Boolean algebra is a field of sets, which in the finite case just means that it consists of $k$-bit numbers for some $k$.

Boolean algebras have found deep applications in set theory, though the relevant Boolean algebras are always infinite. In this application at least, it is certainly possible to think of the values of the Boolean algebra as truth values. One can then obtain a more conventional truth value by using an ultrafilter.