Let's say I have two sets $A = \{ x \ | \ P(x)\}$ and $B = \{y \ | \ Q(y)\}$, how does one define the cartesian product $A \times B$ on these two sets? Is it simply the following?
$$A \times B = \{(x, y) \ | \ P(x) \land Q(y) \}$$
How to define the Cartesian Product on $A = \{ x \ | \ P(x)\}$ and $B = \{y \ | \ Q(y)\}$
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elementary-set-theory
definition
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0Think of the Cartesian product $[0,1],[0,1]$, defining $[0,1]=\{x \in \mathbb{R}: 0 \le x \le 1 \}$. Does this work in that case? (Yes it does) – 2017-01-23
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0$$ \begin{align} z \in (A \times B) &\leftrightarrow \exists x,y: (z=(x,y) \wedge x \in A \wedge y \in B)\\ &\leftrightarrow \exists x,y: (z=(x,y) \wedge P(x) \wedge Q(y)) \\ &\leftrightarrow z \in \{x|P(x)\} \times \{y| Q(y)\} \end{align}$$ – 2017-02-02
1 Answers
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Recall ordered pairs have the following property. $$(a,b)=(x,y)\iff a=x\quad\text{and}\quad b=y$$ That is to say, an ordered pair is well-defined once we know what the first coordinate and second coordinate are.
Given sets $A$ and $B$, the cartesian product, denoted $A\times B$, is the collection of all ordered pairs $(a,b)$, where $a\in A$ and $b\in B$.
In your case, you have $A=\{x:P(x)\}$ and $B=\{y:Q(y)\}$. Therefore $A\times B$ is the collection of all ordered pairs $(a,b)$, where $P(a)$ holds and $Q(b)$ holds. This can indeed be denoted as $$\{(a,b):P(a)\,\wedge\,Q(b)\}$$ as you proposed.