Validity of identities

Task number: 3345

Let \(A\), \(B\) and \(C\) be finite sets.

Determine the validity of the following identities. If they are invalid, transform them to be valid if possible using the smallest possible change (e.g. by introducing appropriate coefficients).

  • Variant

    \( |A \cup B\cup C| = |A| + |B| + |C| - |A \cap B| - |A\cap C| - |B\cap C|- |A \cap B\cap C| \)

  • Variant

    \( |A \setminus B|+ |B \setminus C| + |C \setminus A|= |A \cup B\cup C| - |A \cap B\cap C| \)

  • Variant

    \( |A \cup B\cup C|+ |A \cap B| + |A\cap C| + |B\cap C|= |A \cap B\cap C|+ |A \cup B| + |A\cup C| + |B\cup C| \)

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
Cs translation
Send comment on task by email