# Define the set of all ordered pairs whose first element coincides with $a’$

elementary-set-theorynotation

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.

1. 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 $$a’$$.

2. 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.

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
$$\{c \in C\ \colon \ \exists b \in B, (a', b) = c\}.$$