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?


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:

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


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