How can we see a tuple as a set?

Question

In the book "mathematical logic", it is said that $(a,b)$ is an abbreviation for $\{ \{a,a\},\{a,b \}\}$.

I don't understand this, since firstly, $\{a,a\}=\{a\}$, and secondly, even if this weren't the case, how does $\{ \{a,a\},\{a,b \}\}$ capture the ordered relation of $(a,b)$?

i.e. how do we see a tuple as a set?

See Ordered pair. Correct: ${ a,a }$ is the same of ${ a }$; and the def "works" because it capture the only relevant property of ordered pairs: $(a,b)=(c,d) \text { iff } a=c \text { and } b=d$. — Mauro ALLEGRANZA, Jan 23 '18 at 13:22
$(a,a) = {a,{a, a}} = {a, {a}}$.
${a,{a, b}} = {a, {a, c}} \implies c = b$ — Good Morning Captain, Jan 23 '18 at 13:22

score 1 · Accepted Answer · answered Jan 23 '18 at 13:22

1

$f(a,b) = \{\{a\}, \{a,b\}\}$ captures ordering in the following way: $a$ is contained in the both elements of $f(a,b)$ and $b$ is contained in only one element.

answered Jan 23 '18 at 13:22

Artur Riazanov

905

You need to argue for the case that $a=b$ as well. Also, that indeed $(a,b)=(c,d)$ implies that $a=c$ and $b=d$. Your argument is essentially correct, but far from being sufficient as an actual answer to this question. – Asaf Karagila Jan 23 '18 at 14:11

How can we see a tuple as a set?

1 Answers1