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I am only starting to study Set Theory and I was confused as to why the following is not true.

I was wondering why the Cartesian product of sets whose elements are ordered pairs is not a set of ordered pairs of ordered pairs.

Cartesian Product of sets $A$ and $B$ is defined to be a set $S$ such that $\{(x,y):x \in A\ \text{and}\ y \in B\}$. For set $A := \{\langle a,b\rangle\}$ and set $B := \{c\}$, why is the following not true:

$$A \times B = \{\langle \langle a,b \rangle, c \rangle\}?$$

Since, $\langle a,b \rangle \in A$ and $c\in B$.

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    $\begingroup$ What does $\langle a,b\rangle$ mean? $\endgroup$ – Ennar Sep 2 '17 at 12:53
  • $\begingroup$ it can only take one element per set ... $\endgroup$ – user451844 Sep 2 '17 at 12:58
  • $\begingroup$ @Ennar sorry about that i meant an ordered pair (a,b) $\endgroup$ – chittychitty Sep 2 '17 at 13:04
  • $\begingroup$ Then it is true, who told you it isn't? $\endgroup$ – Ennar Sep 2 '17 at 13:07
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    $\begingroup$ What you have written seems to be true. $\endgroup$ – Teresa Lisbon Sep 2 '17 at 13:10
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I was wondering why the Cartesian product of sets whose elements are ordered pairs is not a set of ordered pairs of ordered pairs.

Let $X\subseteq A\times B$ and $Y\subseteq C\times D$. Then, \begin{align}X\times Y\subseteq (A\times B)\times (C\times D) &= \{(x,y)\mid x\in A\times B,\ y\in C\times D\}\\ &= \{((a,b),(c,d))\mid a\in A,\ b\in B,\ c\in C,\ d\in D\}.\end{align}

What you actually seem to be confused about is why $A\times B\times C \neq (A\times B)\times C$ as sets.

Now, this really depends on how you define Cartesian product of multiple sets. One way actually is just to define $A\times B\times C = (A\times B)\times C$ and then the question is moved to why $$(A\times B)\times C\neq A\times (B\times C).$$

Another way can be $$A_1\times A_2\times A_3 := \{f\colon \{1,2,3\}\to A_1\cup A_2\cup A_3\mid f(i)\in A_i,\ i = 1,2,3\}$$ and then indeed it is not true that $A\times B\times C = (A\times B)\times C.$

You should clarify the definitions you are using.

However, there are natural bijections between sets $$\{(a,b,c)\mid a\in A,\ b\in B,\ c\in C\},\ \{((a,b),c)\mid a\in A,\ b\in B,\ c\in C\}\ \text{and}\ \{(a,(b,c))\mid a\in A,\ b\in B,\ c\in C\}.$$ Can you find them?

Because of those natural bijections, we tend to just write $$A\times B\times C = (A\times B)\times C = A\times (B\times C)$$ even though it is not technically correct.

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