So in class we came across the next definition: Let B be a boolean algebra and E a subset of B, let A be a boolean sub-algebra of B that contains E, we will say that A is generated by E if for every D, D a subalgebra of B, such that E is a subset of D, then A is a subalgebra of D.
And we'll say that A= gen(E).
I find this definition to be very confusing, so I have two main questions:
wouldn't this definition imply that the empty set generates every boolean algebra? Also wouldn't 1 or 0 always be a generator of every subalgebra for every algebra?
why does A have to be a subalgebra of every D?
Thanks in advance for the help.