Difference between $\{a\}$ and $\{\{a\}\}$ in Sets

Question

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.

Answer 1

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!

Answer 2

Think of a box with the letter $A$ and another that has a box with the letter $A$. These are two different boxes - you label them differently!

Answer 3

In layman term, a box that contains a chocolate is different from a box that contains a box that contains a chocolate.

The only element of $P$ is $a$.

The only element of $B$ is $\{a\}$.

Answer 4

P is a set which contains the element "a". Now B is a set which contains the set {a}, or the same, B is a set which contains P. Thus the elements of P and B are not the same (P contains the element a but B contains the set {a} = P)

Answer 5

The sets $P=\{a\}$ and $B=\{\{a\}\}$ are not equal because they do not have the same members.

The only member of the $P$ is $a$ and $a$ is not a member of $B$

On the other hand the only member of $B$ is $\{ a \}$ which is not a member of $P$

You were not sure about the difference between $\{ a \}$ and $a$.

They are not the same because the first one is a set and the second one is an element of that set.