1. Problem A: Set Theory operations and subsets 1.1 Find the subsets of $Q = \{2,4,6\}$. - The subsets of a set with $n$ elements are $2^n$ in number. - Here, $n=3$, so total subsets = $2^3 = 8$. - Subsets are: $\{\}, \{2\}, \{4\}, \{6\}, \{2,4\}, \{2,6\}, \{4,6\}, \{2,4,6\}$. 1.2 Find the proper subsets of $R = \{6,8\}$. - Proper subsets exclude the set itself. - Total subsets of $R$ are $2^2=4$. - Proper subsets are: $\{\}, \{6\}, \{8\}$. 1.3 Find the following sets: - $P' = U - P = \{0,2,4,6,8\}$ - $Q' = U - Q = \{0,1,3,5,7,8,9\}$ - $P \cup R = \{1,3,5,6,7,8,9\}$ - $P \cap R = \{\}$ (no common elements) - $P - Q = P$ (since $P$ and $Q$ are disjoint) 2. Problem B: Seminar qualifications Given: - Total = 200 - $|B|=115$, $|C|=100$, $|A|=70$ - $|B \cap C|=40$, $|C \cap A|=50$, $|B \cap A|=20$ - None qualified = 10 Find $|B \cap C \cap A|$. Use inclusion-exclusion: $$|B \cup C \cup A| = |B| + |C| + |A| - |B \cap C| - |C \cap A| - |B \cap A| + |B \cap C \cap A|$$ Total qualified = $200 - 10 = 190$ So, $$190 = 115 + 100 + 70 - 40 - 50 - 20 + |B \cap C \cap A|$$ $$190 = 285 - 110 + |B \cap C \cap A|$$ $$190 = 175 + |B \cap C \cap A|$$ $$|B \cap C \cap A| = 190 - 175 = 15$$ 3. Problem C: Venn Diagrams sets Sets given: - 1) $U=\{1,2,3,4,5,6,7,8,9\}$, $A=\{2,3,5\}$, $B=\{1,4,6,8\}$ - 2) $U=\{1,2,3,4,5,6,7,8,9\}$, $S=\{1,5,7,8,9\}$, $T=\{5,8\}$ - 3) $A=\{1,2,3,7\}$, $B=\{1,2,4,8\}$, $C=\{3,4,5,6\}$ - 4) $U=\{1,2,3,4,5,6,7,8,9\}$, $X=\{1,2,3\}$, $Y=\{1,2,6,7,8,9\}$, $Z=\{4,5,6,7\}$ (These can be represented in separate Venn diagrams with the universal set and the subsets as given.) 4. Problem D: Music exam results Given: - $|A|=70$ (vocal test passed) - $|B|=75$ (instrument test passed) - $|A \cap B|=55$ 4.1 Find number who passed vocal or instrument test: $$|A \cup B| = |A| + |B| - |A \cap B| = 70 + 75 - 55 = 90$$ 4.2 Find number who passed only one test: - Passed only vocal: $|A| - |A \cap B| = 70 - 55 = 15$ - Passed only instrument: $|B| - |A \cap B| = 75 - 55 = 20$ - Total passed only one test = $15 + 20 = 35$ 4.3 Number who failed both tests: - Total students = 100 - Passed at least one test = 90 - Failed both = $100 - 90 = 10$ 5. Problem E: Bank account holders Given: - $|A|=325$, $|B|=300$, $|C|=260$ - $|A \cap B|=190$, $|B \cap C|=170$, $|A \cap C|=175$ - Others (not A, B, or C) = 30 - Total = 500 Find $|A \cap B \cap C|$. Use inclusion-exclusion: $$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|$$ Since 30 have other accounts, those not in $A \cup B \cup C$ are 30, so: $$|A \cup B \cup C| = 500 - 30 = 470$$ Substitute: $$470 = 325 + 300 + 260 - 190 - 170 - 175 + |A \cap B \cap C|$$ $$470 = 885 - 535 + |A \cap B \cap C|$$ $$470 = 350 + |A \cap B \cap C|$$ $$|A \cap B \cap C| = 470 - 350 = 120$$