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$$A \cap (E_1\cup E_2) = [A\cap E_1]\cup [A\cap E_2\cap E_1^c]$$

Is this a correct relationship? I tried not returned substantiated. Thanks for all.

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Yes it is.

Use:

  • $E_1 \cup E_2 = E_1 \cup (E_2\cap E_1^c)$;
  • $A \cap (B \cup C) = (A \cap B)\cup (A\cap C)$.

I leave the justification to you.

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Hint: on the right hand side, consider $X=A\cap E_1$ , $Y=A\cap E_2$· Continue.

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