Sets question please help :)

IamKyle0

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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.
 
I don't really know how to answer a question like this, so advise will be much appreciated
 
The two singleton elements must be equal and the ... must be equal.
 
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 A={{a1},{a1,a2}} & B={{b1},{b1,b2}}\mathcal{A}=\{\{a_1\},\{a_1,a_2\}\}~\&~\mathcal{B}=\{\{b_1\},\{b_1,b_2\}\} then a1A & b1Ba_1\notin \mathcal{A}~\&~b_1\notin \mathcal{B}.
But {a1}A & {a1,a2}A\{a_1\}\in\mathcal{A}~\&~\{a_1,a_2\}\in\mathcal{A} so if we know that A=B\mathcal{A}=\mathcal{B}.
Then it must be true that {a1}B & {a1,a2}B\{a_1\}\in\mathcal{B}~\&~\{a_1,a_2\}\in\mathcal{B}. Do you follow that?
Knowing that B={{b1},{b1,b2}}\mathcal{B}=\{\{b_1\},\{b_1,b_2\}\} that means {a1}={b1} or {a1}={b1,b2}\{a_1\}=\{b_1\}\text{ or }\{a_1\}=\{b_1,b_2\}
Using both of those can you show that a1=b1 & a2=b2 ?a_1=b_1~\&~a_2=b_2~?
Whoever wrote this problem is getting you ready for the definition of ordered-pairs.
 
If A={{a1},{a1,a2}} & B={{b1},{b1,b2}}\mathcal{A}=\{\{a_1\},\{a_1,a_2\}\}~\&~\mathcal{B}=\{\{b_1\},\{b_1,b_2\}\} then a1A & b1Ba_1\notin \mathcal{A}~\&~b_1\notin \mathcal{B}.
But {a1}A & {a1,a2}A\{a_1\}\in\mathcal{A}~\&~\{a_1,a_2\}\in\mathcal{A} so if we know that A=B\mathcal{A}=\mathcal{B}.
Then it must be true that {a1}B & {a1,a2}B\{a_1\}\in\mathcal{B}~\&~\{a_1,a_2\}\in\mathcal{B}. Do you follow that?
Knowing that B={{b1},{b1,b2}}\mathcal{B}=\{\{b_1\},\{b_1,b_2\}\} that means {a1}={b1} or {a1}={b1,b2}\{a_1\}=\{b_1\}\text{ or }\{a_1\}=\{b_1,b_2\}
Using both of those can you show that a1=b1 & a2=b2 ?a_1=b_1~\&~a_2=b_2~?
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?
 
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