[Measure Theory]How to identify Algebra of sets, Sigma-algebra, there examples and counter examples

Algebra $\mathbb{A}$

This is the collection $\mathcal{A}$ of subsets of X where X is a nonempty set,
satisfying the following conditions.
(i)$\phi\epsilon\mathcal{A}$
(ii)for any set in an algebra, there exists its complements in the algebra.
i.e if $A\epsilon\mathcal{A}$ then $A^c\epsilon\mathcal{A}$
(iii)for any two sets in the algebra $\mathcal{A}$,

there union is also an element of the algebra $\mathcal{A}$,
i.e for $A,B\epsilon\mathcal{A}$ then $A\cup{B}\epsilon\mathcal{A}$.
from the definition above we can conclude that for any two sets in the algebra
then their set difference is also an element of the algebra
i.e $A-B\epsilon\mathcal{A}$ or $A\cap{B^c}\epsilon\mathcal{A}$ and so
is their symmetric difference an element of the algebra.
and we should observe that for a set to be an algebra
then it must be closed under finite unions and intersection.

Examples of Algebra

(1) for any sets X that is non-empty there exists two algebras in X such that
$\mathcal{B}_1=\{\phi,X\}$, and $\mathcal{B}_2=P(X)$,
where $\mathcal{B}_1$ and $\mathcal{B}_2$ are algebras in X,
Now  define an algebra $\mathcal{F}$ in X ,
then $\mathcal{B}_1\subseteq\mathcal{F}\subseteq\mathcal{B}_2$.
(2)if A is a proper subset of a set X and $\mathcal{F}=\{\phi,A,A^c,X\}$. then $\mathcal{F}$ is an algebra in X,
in fact $\mathcal{F}$ is the smallest algebra in X which contains A,
that is if $\mathcal{M}$ is any algebra in X which contains A,
then $\mathcal{F}\subseteq\mathcal{M}$

Counter examples
the subsets of finite sets is countable and hence not an algebra but rather a $\sigma$-algebra.

$\sigma$-algebra

If X is a non-empty set then a collection $\sum$ of subsets of X is called a $\sigma-algebra$
if the following conditions are satisfied:
(i)$\phi\epsilon\sum$
(ii)for any element in the $\sigma-algebra$ there exists its complement in the
$\sigma-algebra$ that is if $A\epsilon\sum$ then there exists $A^c\epsilon\sum$.
(iii) if $(A_n)$ is a sequence of sets in $\sum$ ,
then $\cup^\infty_{n=1}A_n\epsilon\sum$ which is closed under countable
unions and intersections and closed under symmetric difference.

Examples
(1) if X is any non-empty set then there exists two special $\sigma-algebras$ in X,
these are $\mathcal{B}_1=\{\phi,X\}$ and $\mathcal{B}_2=P(X)$.
if $\mathcal{F}$ is any algebra then
$\mathcal{B}_1\subseteq\mathcal{F}\subseteq\mathcal{B}_2$
(2) if X is any uncountable set and $\mathcal{G}$ is a $\sigma-algebra$
then $\mathcal{G}=\{A\epsilon{P(X)},either A or A^c is countabe\}$ 

Counter Example
(1) if X is an infinite set, and $\mathcal{A}$ a collection of all subsets of X which
are finite or have finite complement, then $\mathcal{A}$ is an algebra of sets
which is not a $\sigma-algebra$.
(2) let A be the collection of all finite disjoint unions of all intervals of the
form:$(-\infty,a],(a,b],(b,\infty),\phi,\mathbb{R}$ then A is an algebra
over $\mathbb{R}$ but not a $\sigma$-algebra because union of sets
$\{0,(i-1)/i\}$ for all $i=1$ to $\infty$, then the interval $(0,1)$ does not belong to A.