Let $A$ and $B$ be arbitrary sets with arbitrary elements $a\in A$ and $b\in B$. Then, define the Cartesian product $C=A\times B$, with arbitrary element $c=(a,b)\in A\times B$. Further, take some $a’\in A$.
My goal is to define the set of all ordered pairs whose first element coincides with $a’$: namely, the set $\{(a,b)\in A\times B:a=a’\}$. However, I have two problems.
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First, I am not sure whether the set-builder expression $\{(a,b)\in A\times B:a=a’\}$ is the correct expression to denote the set of all ordered pairs whose first element coincides with $a’$.
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Second, I need to start the definition with $\{c\in C:\dots\}$ rather than with $\{(a,b)\in A\times B:\dots\}$. I have thought of using the set-builder expression $\{c\in C:c_A=a’\}$, but I am not sure whether it is legit to use $c_A$ to denote the first element of the pair $c\in C$.
Thank you all for your help.
Best Answer
Most people would be happy with your expression $\{(a,b) \in A \times B\ \colon \ a=a'\}$ in most contexts. When you want something more formal, perhaps in some contexts in axiomatic set theory, you could write something like
$$\{c \in C\ \colon \ \exists b \in B, (a', b) = c\}.$$