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I am given $2$ sets $A$ and $B$ :

$A = \{1,2,5,6,7\}, B=\{0,4,6,7,9\}$

and two more sets $C = \{0,1,2,6,7,9\}$ and $M = \{0,1,2,3,4,5,6,7,8,9\}$.

I have the following set equation to be solved:

$(A \cap X) \cup (B \cap X^c) = C$

My own thoughts have been to use the law of inverse : $X \cup X^c = \emptyset$.

But I can't use the distributive law for sets stated below:

$A \cup (A \cap B) = (A \cup B) \cap (A \cup C)$

When $X \subseteq M$. So how can I find the X that satisfies $(A \cap X) \cup (B \cap X^c) = C$?

Asaf Karagila
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mire12
  • 95

1 Answers1

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Hint to get started: something is in $C$ iff it's in $A$ and $X$, or in $B$ and not $X$. Since $0$ is in $C$, that means it's in $A$ and $X$ or else it's in $B$ and not in $X$. But it's not in $A$; what does that tell you about whether $0 \in X$?

Patrick Stevens
  • 36,135
  • Thanks! I had to think little but thanks your answer helped me and 0 belongs to the subset X! – mire12 Nov 07 '22 at 11:55