If n(C-D)= 28, n(C union D)= 80 and n(C intersection D)= 35, then find n(D).

Answer 1

To find \( n(D) \), we can use the principle of set operations. We know:

1. \( n(C - D) = 28 \)
2. \( n(C \cup D) = 80 \)
3. \( n(C \cap D) = 35 \)

From set theory, we can express the number of elements in \( C \):

\[ n(C) = n(C - D) + n(C \cap D) \]

Substituting the values we have:

\[ n(C) = 28 + 35 = 63 \]

Next, we can use the formula for the union of two sets:

\[ n(C \cup D) = n(C) + n(D) - n(C \cap D) \]

Now substituting the known values:

\[ 80 = 63 + n(D) - 35 \]

This simplifies to:

\[ 80 = 28 + n(D) \]

Subtracting 28 from both sides gives:

\[ n(D) = 80 - 28 = 52 \]

Thus, the number of elements in set \( D \), \( n(D) \), is:

\[ \boxed{52} \]