Showing a counter example $(A\times B)\times C=A\times (B\times C)$

Question

Showing a counter example

$(A\times B)\times C=A\times (B\times C)$

I think

$A=\{1\}$

$B=\{2\}=C$

Would work but I am not sure...

Answer 1

Let $A=B=C=\{\emptyset\}$. Then $$ (A\times B)\times C=\{((\emptyset,\emptyset),\emptyset)\}$$ whereas $$ A\times (B\times C)=\{(\emptyset,(\emptyset,\emptyset))\}$$ This looks different. If we use the definition $(x,y):=\{\{x\},\{x,y\}\}$, we see that indeed $$\begin{align} (\emptyset,\emptyset)&=\{\{\emptyset\}\},&\text{ hence}\\ (A\times B)\times C& = \{ \{ \{\{\{\emptyset\}\}\},\{ \emptyset ,\{\{\emptyset\}\}\}\} \}&\text{and}\\ A\times (B\times C)&=\{ \{ \{\emptyset\},\{\{\emptyset\},\{\{\emptyset\}\} \} \} \} \end{align}$$ which are truely different sets.