Difference between $\{a\}$ and $\{\{a\}\}$ in Sets I am presently a 11th grader. I came across a question which was to answered of True or False in the Sets chapter. Let me just text down the complete question for more clarity. > The sets $P=\\{a\\}$ and $B=\\{\\{a\\}\\}$ are equal. $\quad$[True/False] I got the answer wrong by opting for True option. However, there is no explanation given about the question. Please help me to resolve this problem.

If a set is given as a list of its elements, the elements are what you get by stripping off **one** pair of curly brackets. Examples:

* the elements of $\\{1,2,3\\}$ are $1,2,3$: these are three numbers;
* the elements of $\\{1,2,\\{3\\}\\}$ are $1,2,\\{3\\}$: these are two numbers and a set: the latter set has one element $3$, which is a number;
* the elements of $\\{1,\\{2,3\\}\\}$ are $1,\\{2,3\\}$: these are a number and a set: the latter set has two elements $2,3$, both of which are numbers;
* $\\{\\{1,2,3\\}\\}$ has only one element, the set $\\{1,2,3\\}$.



I hope you can see that the elements are different in all four cases, so the sets are different. For your example,

* $\\{a\\}$ has one element which is a letter;
* $\\{\\{a\\}\\}$ has one element which is a set.



A letter is not the same as a set, so the elements are different, so $\\{a\\}$ is different from $\\{\\{a\\}\\}$.

Hope this helps!