Is it true "Every Boolean algebra is an algebra of sets, for any given set X"

Question

I have confused between these two notions, please help

Every Boolean algebra is an algebra of sets, for any given set $X$, and the converse is false.

Answer 1

What the highlighted proposition is stating amounts to: (I) AND (II)

(I) $\implies:\;\;$It is true that IF B is a Boolean algebra, THEN B is (isomorphic to) an algebra of sets.

(II) $\not\Longleftarrow:\;\;$ But it is not true that every algebra of sets, is (isomorphic to) a Boolean algebra. That is, the converse of (I) is not true.

Is that what you're confused about?

Any Boolean algebra is (isomorphic to) an algebra of sets. Simple enough: there is no Boolean algebra that fails to be isomorphic to an algebra of sets. (I.e., if it fails to be isomorphic to an algebra of sets, then it certainly cannot be a Boolean algebra.)

But surely we can find of an algebra of sets that fails to be (isomorphic to) a Boolean Algebra.