1. **Set Theory Problem (A)**: Given sets $A=\{1,2,4,5\}$, $B=\{2,3,5,6\}$, $C=\{4,5,6,7\}$, verify: (a) $A - (B \cap C) = (A - B) \cup (A - C)$ - First find $B \cap C = \{5,6\}$. - Then $A - (B \cap C) = \{1,2,4,5\} - \{5,6\} = \{1,2,4\}$. - Next, $A - B = \{1,4\}$ and $A - C = \{1,2\}$. - Their union $(A - B) \cup (A - C) = \{1,2,4\}$. - Both sides equal $\{1,2,4\}$, so the equality holds. (b) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ - Find $B \cup C = \{2,3,4,5,6,7\}$. - Then $A \cap (B \cup C) = \{1,2,4,5\} \cap \{2,3,4,5,6,7\} = \{2,4,5\}$. - Next, $A \cap B = \{2,5\}$ and $A \cap C = \{4,5\}$. - Their union $(A \cap B) \cup (A \cap C) = \{2,4,5\}$. - Both sides equal $\{2,4,5\}$, so the equality holds. 2. **Set Theory Problem (A) with intervals**: Given $A' = \{x \in \mathbb{R} | 1 \leq x^2 \leq 10\}$, $A = \{x \in A' | -2 < x \leq 3\}$, $B = \{x \in A' | 1 \leq x^3 \leq 10\}$. (1) Find $A \cap B$. - $A'$ means $x^2$ between 1 and 10, so $x \in [-\sqrt{10}, -1] \cup [1, \sqrt{10}]$. - $A$ restricts $x$ to $(-2,3]$ intersected with $A'$, so $A = (-2,-1] \cup [1,3]$. - $B$ requires $1 \leq x^3 \leq 10$, so $x \in [1, \sqrt[3]{10}]$. - Since $\sqrt[3]{10} \approx 2.15$, $B = [1, 2.15]$. - Then $A \cap B = [1, 2.15]$. (2) Find $A' - (A \cap B)$. - $A' = [-\sqrt{10}, -1] \cup [1, \sqrt{10}]$. - $A \cap B = [1, 2.15]$. - So $A' - (A \cap B) = [-\sqrt{10}, -1] \cup (2.15, \sqrt{10}]$. (3) Find $(A - B) \cup (B - A)$. - $A - B = (-2,-1] \cup (2.15,3]$ (since $A$ includes $(-2,-1]$ and $[1,3]$, removing $B=[1,2.15]$ leaves $[2.15,3]$). - $B - A = \emptyset$ (since $B \subseteq A$). - So union is $(-2,-1] \cup (2.15,3]$. (4) Find $n(A \cap B)$. - Since $A$ and $B$ are intervals, $n(A \cap B)$ is infinite (uncountable). 3. **Logarithm Properties**: - Property 1: $\log_b(xy) = \log_b x + \log_b y$ - Property 2: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$ - Property 3: $\log_b (x^k) = k \log_b x$ Show that if $x > y > 0$, then $\log x > \log y$ (assuming base $b > 1$). - Since $x > y > 0$ and $\log_b$ is increasing for $b > 1$, it follows $\log_b x > \log_b y$. 4. **Boolean Algebra**: Given Boolean expression $X_1 + X_2 X_1 + (X_2 X_3') (X_1 + X_2)' + (X_1' X_3)$, find sum of products canonical form. - Apply Boolean algebra rules and distributive laws to simplify and express as sum of products. 5. **Matrix Inversion and Verification**: Given matrix $A$, find $A^{-1}$ and verify $A A^{-1} = I$. - Use formula for inverse or row operations. - Multiply $A$ and $A^{-1}$ to check if product is identity matrix. 6. **Probability Problem**: Factory A, B, C produce 25%, 35%, 40% of balls with defective rates 5%, 4%, 2% respectively. Find probability that a randomly selected defective ball is from factory B. - Use Bayes' theorem: $$P(B|D) = \frac{P(D|B)P(B)}{P(D|A)P(A) + P(D|B)P(B) + P(D|C)P(C)} = \frac{0.04 \times 0.35}{0.05 \times 0.25 + 0.04 \times 0.35 + 0.02 \times 0.40} = \frac{0.014}{0.0125 + 0.014 + 0.008} = \frac{0.014}{0.0345} \approx 0.4058$$ 7. **Combinatorics**: Number of words formed from letters of "ALLAHABAD" with vowels in even positions or 2 L's not together can be found using permutations with restrictions. 8. **Miscellaneous**: - Number of elements in power set of empty set $P(\emptyset)$ is 1. - Function $f: \mathbb{N} \to \mathbb{N}, f(x) = x + 1$ has domain and codomain $\mathbb{N}$. - Relation $f(x) = x - 1$ is not a function from $\mathbb{N}$ to $\mathbb{N}$ because $f(0) = -1 \notin \mathbb{N}$. - If domain and range are same, relation may be reflexive. - Partial order relation is reflexive, antisymmetric, and transitive. - Equivalence relation is reflexive, symmetric, and transitive. - Conditional probability is probability of second event given first event. - Mutually exclusive events cannot occur together. - Updating theory refers to Bayesian updating.