Given two sets A and B, which one of the following is equivalent to A complement union B complement?

Steps to Solve

1. Understand the Complement Operation The complement of a set $A$, denoted $A^c$, includes all elements not in $A$. Therefore, if we start with the expression $A^c \cup B^c$, we need to find what this means in terms of A and B.

2. Apply De Morgan's Laws According to De Morgan's laws, the union of the complements can be expressed as the complement of the intersection: $$ A^c \cup B^c = (A \cap B)^c $$

3. Analyzing Set Operations We need to find an equivalent expression among the options given. Let's derive expressions based on $(A \cap B)^c$:

   • This consists of everything outside both A and B, including elements not in A or not in B.

4. Evaluate the Given Options

   • $A - B$: Elements in A that are not in B.
   • $\emptyset$: The empty set, which doesn't represent our derived expression.
   • $A^c \cap B^c$: Elements that are outside both A and B, which is consistent with $(A \cap B)^c$.
   • $B - A$: Elements in B that are not in A.

5. Identify the Correct Answer From our analysis, the correct representation of $A^c \cup B^c$ is given by $A^c \cap B^c$.