Question

If A, B, and C are any three non-empty sets. Prove that ( overline{A cap B cap C} = overline{A} cup overline{B} cup overline{C} )

a. Set union

b. Set intersection

c. Set difference

d. Set complement

Answer 1

The option is d. Set complement. The equality
`\(\overline{A \cap B \cap C} = \overline{A} \cup \overline{B} \cup \overline{C}\)` is established by applying De Morgan's Law, stating that the complement of the intersection of sets A, B, and C is equivalent to the union of their individual complements.

Step-by-step explanation:

To prove the given set equality
`\(\overline{A \cap B \cap C} = \overline{A} \cup \overline{B} \cup \overline{C}\),` we can use set complement properties. Starting with the left side,
`\(\overline{A \cap B \cap C}\),` this represents the complement of the intersection of sets A, B, and C.

According to De Morgan's Law, the complement of an intersection is equivalent to the union of complements. Therefore,
`\(\overline{A \cap B \cap C} = \overline{A} \cup \overline{B} \cup \overline{C}\).`

This set equality can be understood by considering the elements not in the intersection of A, B, and C, which is the same as the union of elements not in A, not in B, and not in C. In set notation,
`\(\overline{A \cap B \cap C} = \overline{A} \cup \overline{B} \cup \overline{C}\)` is a concise way of expressing this relationship.

Hence, the correct choice among the provided options is d. Set complement, as the equality involves the complement of the intersection on one side and the union of complements on the other side, aligning with the properties of set complements.