To identify the theorems and/or identities that justify each step, let's carefully analyze the steps and match them to standard set-theoretic principles:

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1. Step:
   $N(A \cap B) = N((A \cap B) \cap U)$
   Justification: Intersection with the universe
   The identity $S = S \cap U$, where $U$ is the universal set, holds because intersecting a set with $U$ doesn't change the set.

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2. Step:
   $N((A \cap B) \cap U) = N(U \cap (A \cap B))$
   Justification: Commutativity of intersection
   The order of intersection does not matter: $S \cap T = T \cap S$.

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3. Step:
   $N(U \cap (A \cap B)) = N(U \cap (A \cap B)^c)^c$
   Justification: Complement rule
   The complement of a complement restores the original set: $S = (S^c)^c$.

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4. Step:
   $N(U \cap (A \cap B)^c)^c = N(U - (A \cap B)^c)$
   Justification: Set difference as intersection with complement
   $S - T = S \cap T^c$, which matches the transformation here.

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5. Step:
   $N(U - (A \cap B)^c) = N(U) - N((A \cap B)^c)$
   Justification: Principle of inclusion-exclusion for complements
   The number of elements in $S - T$ equals the total elements in $S$ minus the elements in $T$.

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6. Step:
   $N((A \cap B)^c) = N(A^c \cup B^c)$
   Justification: De Morgan's laws
   $(A \cap B)^c = A^c \cup B^c$, which relates complements of intersections to unions of complements.

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7. Step:
   $N(A^c \cup B^c) = N(A^c) + N(B^c) - N(A^c \cap B^c)$
   Justification: Principle of inclusion-exclusion for unions
   The cardinality of a union is the sum of individual cardinalities minus the overlap.

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Summary of Justifications:

1. Intersection with the universe
2. Commutativity of intersection
3. Complement rule
4. Set difference as intersection with complement
5. Inclusion-exclusion for complements
6. De Morgan's laws
7. Inclusion-exclusion for unions

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Would you like me to elaborate on any of these justifications? Here are 5 related questions:

1. How do De Morgan's laws apply to set complements in real-world scenarios?
2. Can you explain inclusion-exclusion principles with a practical example?
3. How does the concept of the universal set simplify operations in set theory?
4. What is the relationship between set differences and intersections with complements?
5. Why is commutativity essential in set operations?

Tip: When working with set theory, drawing Venn diagrams often clarifies the operations and helps visualize relationships.