Sets question please help :)

[IamKyle0]

Consider the sets A = {{a1}, {a1, a2}} and B = {{b1}, {b1, b2}}. Prove that for every a1, a2, b1, b2, A = B implies a1 = b1 and a2 = b2.

 

[IamKyle0]

I don't really know how to answer a question like this, so advise will be much appreciated

 

[Steven G]

The two singleton elements must be equal and the ... must be equal.

 

[pka]

Consider the sets A = {{a1}, {a1, a2}} and B = {{b1}, {b1, b2}}. Prove that for every a1, a2, b1, b2, A = B implies a1 = b1 and a2 = b2.

If [imath]\mathcal{A}=\{\{a_1\},\{a_1,a_2\}\}~\&~\mathcal{B}=\{\{b_1\},\{b_1,b_2\}\}[/imath] then [imath]a_1\notin \mathcal{A}~\&~b_1\notin \mathcal{B}[/imath].
But [imath]\{a_1\}\in\mathcal{A}~\&~\{a_1,a_2\}\in\mathcal{A}[/imath] so if we know that [imath]\mathcal{A}=\mathcal{B}[/imath].
Then it must be true that [imath]\{a_1\}\in\mathcal{B}~\&~\{a_1,a_2\}\in\mathcal{B}[/imath]. Do you follow that?
Knowing that [imath]\mathcal{B}=\{\{b_1\},\{b_1,b_2\}\}[/imath] that means [imath]\{a_1\}=\{b_1\}\text{ or }\{a_1\}=\{b_1,b_2\}[/imath]
Using both of those can you show that [imath]a_1=b_1~\&~a_2=b_2~?[/imath]
Whoever wrote this problem is getting you ready for the definition of ordered-pairs.

 

[IamKyle0]

If [imath]\mathcal{A}=\{\{a_1\},\{a_1,a_2\}\}~\&~\mathcal{B}=\{\{b_1\},\{b_1,b_2\}\}[/imath] then [imath]a_1\notin \mathcal{A}~\&~b_1\notin \mathcal{B}[/imath].
But [imath]\{a_1\}\in\mathcal{A}~\&~\{a_1,a_2\}\in\mathcal{A}[/imath] so if we know that [imath]\mathcal{A}=\mathcal{B}[/imath].
Then it must be true that [imath]\{a_1\}\in\mathcal{B}~\&~\{a_1,a_2\}\in\mathcal{B}[/imath]. Do you follow that?
Knowing that [imath]\mathcal{B}=\{\{b_1\},\{b_1,b_2\}\}[/imath] that means [imath]\{a_1\}=\{b_1\}\text{ or }\{a_1\}=\{b_1,b_2\}[/imath]
Using both of those can you show that [imath]a_1=b_1~\&~a_2=b_2~?[/imath]
Whoever wrote this problem is getting you ready for the definition of ordered-pairs.

Oh I think I understand. So for line 4, you put {a1}={b1,b2} but it should be {a1, a2}={b1, b2} correct?