Let us consider the following set:

$A=\{(x, y, z) \in \Bbb{R}\times\Bbb{R}\times\Bbb{R} : ax+by+c=0,z=0 \},c\neq 0$ and $B=\{(x, y, z) \in \Bbb{R}\times\Bbb{R} \times\Bbb{R} : ax+by=0,z=0\}$. Then $A, B$ are two infinite sets. How to determine the cardinality of the set $A-B$?


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I am assuming $a$, $b$, and $c$ are fixed real numbers.

If $(x,y,z)\in B$, then $z=0$ and $ax+by=0$. But then $ax+by\neq -c$, since $c\neq 0$, so $(x,y,z)\not\in A$. Therefore, $A-B=A$.

This is really just a proof that two parallel (but distinct) lines have no points of intersection!


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