Collection of Maths Problems

Validity of identities

Task number: 3345

• Variant

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

• Solution

  We draw a Venn diagram and count how many times elements from all possible subsets are counted on the left and right sides of the identity.

  We determine that all elements of the union except the elements \(A \cap B\cap C\) are counted once on both sides of the identity.

  The elements \(A \cap B\cap C\) are counted once on the left, whereas on the right they are counted once in each of the sets, i.e. altogether \(1+1+1-1-1-1-1=-1\) time.

• Answer

  The identity is invalid. The following is valid, for example: \( |A \cup B\cup C| = |A| + |B| + |C| - |A \cap B| - |A\cap C| - |B\cap C|+ |A \cap B\cap C| \)

• Variant

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

• Solution

  From a Venn diagram we can see that in fact

  \((A \setminus B) \cup (B \setminus C) \cup (C \setminus A) = (A \cup B\cup C) \setminus (A \cap B\cap C)\).

• Answer

  The identity \( |A \setminus B|+ |B \setminus C| + |C \setminus A|= |A \cup B\cup C| - |A \cap B\cap C| \) is valid.

• Variant

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

• Solution

  We draw a Venn diagram and count how many times elements from all possible subsets are counted on the left and right sides of the identity.

  On the left, elements belonging to only one of the sets are counted once, elements belonging to two sets are counted twice and elements in the common intersection are counted four times.

  On the right, elements belonging to only one set are counted twice, elements belonging to two sets are counted three times and elements in the common intersection are counted four times.

  We can determine appropriate coefficients with a bit of thought. In more complicated examples we could set up a series of linear equations.

• Answer

  The identity is invalid. The following is correct, for example:

  \( 2|A \cup B\cup C|+ |A \cap B| + |A\cap C| + |B\cap C|= 2|A \cap B\cap C|+ |A \cup B| + |A\cup C| + |B\cup C| \)