Are the following statements true? (1) a∈{{a},{a,b}} and (2) b∈{{a},{a,b}} true?

[john-ice2023]

TL;DR Summary: Look deep into nature, and then you will understand everything better. Albert Einstein.

I am new to set theory. I got confused about above questions.
For Q(1), I have two solutions,
(a) because a is not the element of set {{a},{a,b}}, so a∈{{a},{a,b}} is False.
(b) because {a}∈{{a},{a,b}} and a∈{a}, therefore a∈{{a},{a,b}} is True.
which one is correct?
Thanks! John.

(MENTOR note): looks like homework so moved to a HW forum but template is missing.

 

[FactChecker]

This looks like a homework problem. We are only allowed to give hints after we see your work. There is a format for homework questions.
In your work, you should state exactly what the members of the set {{a}, {a,b}} are.

 

[john-ice2023]

This is not homework. I saw it on the Internet. I just want to learn something new.

 

[phinds]

This is not homework. I saw it on the Internet. I just want to learn something new.

Yes, but the problem is that we have no way of confirming that so when it even LOOKS like a homework problem, we do ask that the poster show some effort on their own rather than just asking for an answer.

 

[jedishrfu]

Its pretty straight forward by understanding the set notation and definition
- a is an element
- {a} is a set containing the element a
- {{a}} is a set containing the set {a} which contains the element a

so:

- is a and element of the set {a}?

- is a an element of the set {{a}}?

- is {a} and element of the set {{a}}?

What do you think the answers are?

 

[Gavran]

$a$ belongs to $\{ a \}$, but $a$ is not equal to $\{ a \}$.

 

[jedishrfu]

Calling @john-ice2023 it'd be nice if you responded to the thread you created sometime soon.

 

[PeroK]

TL;DR Summary: Look deep into nature, and then you will understand everything better. Albert Einstein.

I am new to set theory. I got confused about above questions.
For Q(1), I have two solutions,
(a) because a is not the element of set {{a},{a,b}}, so a∈{{a},{a,b}} is False.
(b) because {a}∈{{a},{a,b}} and a∈{a}, therefore a∈{{a},{a,b}} is True.
which one is correct?
Thanks! John.

(MENTOR note): looks like homework so moved to a HW forum but template is missing.

Here's an analogy. The Rugby World Cup is on at the moment. We have a set of twenty teams in the competition. Each team is a set of about 30 players. So, if $a$ is a player, then $a$ may be a member of one of the teams. But $a$ is a not a team. So, $a$ is not a member of the set of teams.

Now, having a set with a single member may seem to cloud the issue. If one of these countries had so few rugby players, that their team consisted of a single player, then in everyday language that player is the team! But, mathematics isn't about everyday language. Mathematics is about precise, formal definitions. And, in terms of set theory, there is still a clear distinction between a player and a team consisting of a single player!

 

[WWGD]

IIRC, {{a},{a, b }} is a way of defining the ordered pair (a,b).