Why in $\mathbb{Z}_{20}$, $\langle 8,14 \rangle=\{0,2,4,\ldots,18\}=\langle 2\rangle$? Why in $\mathbb{Z}_{20}$, $\langle 8,14 \rangle=\{0,2,4,\ldots,18\}=\langle 2\rangle$?
I don't understand the notation, what does $\langle 8,14 \rangle$ mean? 
I know the definition $\langle a \rangle$ means the smallest subgroup of a group G containing a, but I can't figure out what $\langle a,b \rangle$ mean. 
also, why $\langle 8,14 \rangle  = \langle 2 \rangle$?
I know $\text{gcd}(8,14)=2$, but what do $\{0,2,4,...,18\}$ stand for?
 A: First of all $\langle 8,14 \rangle$ means the subgroup generated by the elements $8$ and $14$ in $\mathbb{Z}_{20}$. In other words it's the smallest subgroup that contains both $8$ and $14$. 
For more general case you can prove that $\langle a_1, a_2, ..., a_n \rangle = \langle \operatorname{gcd}(a_1, a_2, ..., a_n) \rangle$. Try to prove that each set is included in the other one and hence the equality. This shouldn't be so hard, as after all you need to prove that only the generators of the subgroup are included in the other subgroup.
Finally, $\{0,2,...,18\}$ are just the elements of the subgroup listed as a set.
A: The notation $\langle 8,14\rangle$ means the smallest subgroup that contains both $8$ and $14$, i.e. the smallest set of congruence classes modulo $20$ that contains $8$ and $14$ and is closed under the group operation and under inversion. This is called the subgroup generated by $8$ and $14.$ When there are only finitely many elements in the group, as in this case, if a set is closed under the group's binary operation then it is also closed under inversion. Notice that $14-8-8 = -2,$ so $-2$ must be included and so $+2$ must be included. And the group generated by $2$ contains all ten of the even numbers among the $20$ numbers involved. Since it contains all the even numbers, it contains $8$ and $14.$
The notation $\{0,2,4,\ldots,18\}$ just means the set whose members are those listed, i.e. $0,2,4,6,8,10,12,14,16,18.$
A: $\{0, 2, 4, \ldots, 18\}$ is just the set containing those elements.  This is a very common notation in many branches of mathematics. 
"what does $\langle 8,14 \rangle$ mean?"  You answered that yourself, it's the smallest subgroup containing those elements.  $\{8, 14\}$ by itself is not a subgroup since it does not contain $8 + 14 = 2$ (note $2$ not $22$ since we are in $\Bbb{Z}_{20}$.  For a small example such as this, you can figure out what the smallest group containing this set is just by playing around.  $8 + 14 = 2$ tells us that if $8$ and $14$ are in a subgroup then $2$ must be.  Once you have $2$ then you must have $4 = 2 + 2$, $6 = 2 + 2 + 2$, and all of the other even numbers.  However, you are not forced to include any odd numbers.  So,whether you start from $\langle 2 \rangle$ or $\langle 8,14 \rangle$, you end up with $\{0, 2, 4, . . ., 18\}$.  
Be careful with odd and even arguments.  It works here since $20$ is even.  In $\Bbb{Z}_{19}$, $\langle 2 \rangle$ would be the whole group.  
A: Well, $\langle8,14\rangle$ is the subgroup generated by $14$ and $8$. And, since $2=2\times8-14$, $2\in\langle8,14\rangle$. Therefore $\langle2\rangle\subset\langle8,14\rangle$. On the other hand, since $8,14\in\langle2\rangle$, $\langle8,14\rangle\subset\langle2\rangle$. Therefore, the groups are equal.
Besides $\langle2\rangle=\{0,2,4,\ldots,18\}$.
