Exercise
35
[1Y9] Let be given \(a,b,x,y\).
Show that in the hypothesis
\[ \{ a,b\} = \{ x,y\} \]you have that
\[ (a= b)\iff (x= y) \iff a=b=x=y \quad . \]In particular, you deduce that if
\[ \{ a\} = \{ x,y\} \]then \(a=x=y\).
Then show that if we assume that the four elements \(a,b,x,y\) are not all the same, then we have
\[ \{ a,b\} = \{ x,y\} \]if and only if \(a=x∧ b=y\) or \(a=y∧ b=x\).
To show the above be as precise as possible: use the axiom of extensionality [1Y8], the axiom of pairing [1Y3] and the tautulogies shown in the previous section (or other elementary logical relationships).
Solution
1