To address the problem of writing an expression for "all of the elements which are in sets A and B (not just A or just B)," we need to understand the set operations involved.
1. Union ( [tex]\( A \cup B \)[/tex] ): This operation represents all elements that are in set A, set B, or in both.
2. Intersection ( [tex]\( A \cap B \)[/tex] ): This operation includes all elements that are in both sets A and B.
3. Complementor Cur' operation [tex]\( P(A \cap B) \)[/tex]: In standard notation, [tex]\( P(A \cap B) \)[/tex] can refer to the power set of the intersection of A and B, though it is often misunderstood as simply the intersection. For clarity, we will avoid this notation due to its potential confusion.
4. Complement ( [tex]\( \left(A' \cup B' \right) \)[/tex]): This operation involves all elements that are not in set A or set B (the complement of the union of A and B).
Given the definitions, the problem statement specifies "all of the elements which are in sets A and B (not just A or just B)." This requirement is directly addressed by the intersection of sets A and B. The intersection contains precisely those elements that are common to both sets A and B.
Thus, the correct expression that matches the given statement is:
[tex]\[ (A \cap B) \][/tex]
This corresponds to the expression that includes all elements common to both sets, without any elements that are solely in either set A or set B.
Therefore, the correct answer is:
[tex]\[ \boxed{(A \cap B)} \][/tex]
