Final answer:
The steps in the derivation involve using identities and theorems:
- De Morgan's law: N(U) - N(Ac U Bc)
- Difference rule: N(U) - N((A n B))
- Double complement law: N(U - (((A n B)c)c))
- Idempotent law: N(U n ((A n B)c)) ( in question it contains typo or mistake).
- Identity law: N(A n B) = N(U n (A n B))
- Inclusion/exclusion rule: N(U) - (N(Ac) + N(Bc) - N(Ac n Bc))
- Set difference law: N(U) - N((A n B))
Step-by-step explanation:
The derivation provided involves manipulating a set expression to find the cardinality, N, of the intersection of two sets, A and B.
Each step in the given derivation uses one of the basic principles of set theory.
Here's a justification of each step:
- N(A n B) = N(U n (A n B)): This step uses the identity law, which states that the intersection of any set with the universal set is the set itself.
- N(U n ((A n B)c)): This step is not immediately clear from context as it seems to be based on a typo or mistake.
- N(U - (((A n B)c)c)): Here the double complement law is applied, which says that the complement of the complement of a set is the set itself.
- N(U - (A n B)): The complement law is used to rewrite the expression such that the complement of the intersection of A and B is equivalent to the intersection of the complements of A and B.
- N(U) - N((A n B)): This uses the set difference law, which allows us to write the cardinality of a set difference as the difference of the cardinalities.
- N(U) - N(Ac U Bc): The De Morgan's law is utilized to convert the complement of the intersection into the union of the complements.
- N(U) - (N(Ac) + N(Bc) - N(Ac n Bc)): The inclusion-exclusion principle is applied to express the cardinality of the union of two sets as the sum of the cardinalities of the individual sets minus the cardinality of their intersection.