suppose c is a subset of d and d is a subset of c. if n(c) = 5, find n (d).
To find the cardinality of set d, given that set c is a subset of d and d is a subset of c, we need to understand a concept called the inclusion-exclusion principle.
The inclusion-exclusion principle states that for two sets A and B, the cardinality of the union of A and B is equal to the sum of the cardinalities of A and B minus the cardinality of their intersection.
In this case, since c is a subset of d and d is a subset of c, it means that c and d are equal, or in other words, c = d. So the intersection of c and d is also equal to c.
Applying the inclusion-exclusion principle, we have:
n(d U c) = n(d) + n(c) - n(d ∩ c)
Since c = d, we can substitute c for d in the equation:
n(c U c) = n(c) + n(c) - n(c ∩ c)
Since c ∩ c is just c itself, we can simplify further:
n(c U c) = 2 * n(c) - n(c)
Given that n(c) = 5, we can substitute it into the equation:
n(c U c) = 2 * 5 - 5
n(c U c) = 10 - 5
n(c U c) = 5
Therefore, the cardinality of set d is 5.