0
$\begingroup$

I'm trying to express the complement $X^C$ of the following set:

$$X = \{ z \mid (b+z) \in A, b \in B \}$$

From the definition of the complement $A^C = \{ a \mid a \notin A \}$, I am not sure which of the following is the complement $X^C$:

  • $\{ z \mid (b+z) \notin A, b \notin B \}$
  • $\{ z \mid (b+z) \notin A, b \in B \}$
  • $\{ z \mid (b+z) \in A, b \notin B \}$

Could someone make this clear to me?

$\endgroup$
0
$\begingroup$

Your $x$ contains all $z$ for which $b+z \in A$ and for which $b \in B$. So, the complement would be all $z$ for which that is not true, i.e. for which $b+z \not \in A$ or for which $b \not \in B$. Which translates to:

$$X^C = \{ z | (b+z) \not \in A \lor b \not \in B \}$$

... which is neither one of your options, unless you take the union of all the ones you have:

$$X^C = \{ z | (b+z) \not \in A, b \not \in B \} \cup \{ z | (b+z) \not \in A, b \in B \} \cup \{ z | (b+z) \in A, b \not \in B \}$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.