Give me a nice question on determining and finding the number of subsets in a set .a tricky one only onw with also the solution

Question

GPT-4o mini · Answer

**Question:** Consider the set $ S = \{ a, b, c, d, e \} $ which contains 5 elements. You want to form subsets of $ S $ that contain at least one element from the set $ T = \{ a, b \} $ and at least one element from the set $ U = \{ c, d \} $. How many such subsets can be formed? **Solution:** 1. **Total Subsets**: The total number of subsets of a set with $ n $ elements is given by $ 2^n $. For the set $ S $ with 5 elements, the total number of subsets is: $$ 2^5 = 32 $$ 2. **Calculate subsets not containing elements from T**: First, calculate the number of subsets that **do not** contain any elements from $ T $. If we exclude $ \{ a, b \} $, we are left with $ \{ c, d, e \} $, which has 3 elements. The number of subsets of this remaining set is: $$ 2^3 = 8 $$ 3. **Calculate subsets not containing elements from U**: Now, calculate the number of subsets that **do not** contain any elements from $ U $. If we exclude $ \{ c, d \} $, we are left with $ \{ a, b, e \} $, which also has 3 elements. The number of subsets of this remaining set is: $$ 2^3 = 8 $$ 4. **Calculate subsets that do not contain elements from both T and U**: Next, find the subsets that do not contain elements from both $ T $ and $ U $. This means we are only considering the element $ e $, which gives us: $$ 2^1 = 2 \quad \text{(the subsets are $ \emptyset $ and $ \{ e \} $)} $$ 5. **Apply the Principle of Inclusion-Exclusion**: Now, we can apply the principle of inclusion-exclusion to find the number of subsets that do not meet our requirements (i.e., do not contain at least one element from $ T $ and at least one element from $ U $): $$ \text{Not containing } T \cup U = (\text{Not } T) + (\text{Not } U) - (\text{Not both } T \text{ and } U) $$ Thus, we have: $$ 8 + 8 - 2 = 14 $$ 6. **Calculate valid subsets**: We can now calculate the number of valid subsets that satisfy our original condition (containing at least one element from $ T $ and at least one element from $ U $): $$ \text{Total subsets} - \text{Not containing } T \cup U = 32 - 14 = 18 $$ Thus, the total number of subsets of $ S $ that contain at least one element from $ T $ and at least one element from $ U $ is **18**.