Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

enter image description here

I know that (a,b) = (c,d) if a = c and b = d, but I have no idea what to do here. I assume I'm supposed to show that {{a},{a,b}} = {{c},{c,d}} if a = c and b = d, but how do I verify that using the sets?

share|improve this question
    
Not "if" but "iff". –  Hagen von Eitzen Nov 13 '13 at 6:58
    
@Foxic You're asked to verify if $(a,b)=(c,d)\iff a=c\land c=d$, according to each definition of ordered pair. –  Git Gud Nov 13 '13 at 6:59
    
How can I do that? –  Foxic Nov 13 '13 at 7:01
    
@Foxic If you realise that that's what you need to do,then please edit your question. As it stands, it looks like you're asking what you need to do. –  Git Gud Nov 13 '13 at 7:02
    
I edited it, I really need help here –  Foxic Nov 13 '13 at 7:20

1 Answer 1

up vote 1 down vote accepted

The first definition doesn't work. Just take any example with $x\neq y$ to generate two different ordered pairs that equal $\{x,y\}$.

Let $a,b,c,d$ be such that $\{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}$. Either $a=b$ or $a\neq b$.

If $a=b$, then $\{\{a\}\}=\{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}$. Thus, due to the cardinality of the sets, $\{a\}=\{c\}$ and $\{a\}=\{c,d\}$, it follows that $a=b=c=d$.

If $a\neq b$, then because $\{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}$, again due to cardinality, it follows that $\{a\}=\{c\}$ and $\{a,b\}=\{c,d\}$, (this needs to be justified a little bit better, consider all possibilities). So $\color{blue}{a=c}$ and $b\in \{a,b\}=\{c,d\}=\{a,d\}$, hence $\color{blue}{b=d}$.

The other direction is trivial.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.