# Defining sets using pairs, check if definition satisfies the pair correctness property - Kuratowski ordered pair

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?

-
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

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 $\color{blue}{b=d}$.