Questions & Answers: "Sets"

First, we solve the quadratic equation for set $A$: $x^2 - 5x + 6 = 0 \implies (x-2)(x-3) = 0$, so $A = \{2, 3\}$. For set $B$, $x^2 = 9 \implies x = \pm 3$, so $B = \{-3, 3\}$. The symmetric difference $A \Delta B$ is defined as $(A \cup B) - (A \cap B)$. Here $A \cup B = \{2, 3, -3\}$ and $A \cap B = \{3\}$. Thus, $A \Delta B = \{2, -3\}$. The number of elements in $A \Delta B$ is $n = 2$. The number of elements in the power set $P(S)$ is given by $2^n$. Therefore, $n(P(A \Delta B)) = 2^2 = 4$. Option 1 is incorrect because it is the cardinality of the symmetric difference itself, not its power set. Options 3 and 4 are incorrect because they use wrong values for $n$ in the $2^n$ formula.

Let $U$ be the universal set of students, $A$ be the set of students taking apple juice, and $O$ be the set of students taking orange juice. We are given $n(U) = 400$, $n(A) = 100$, $n(O) = 150$, and $n(A \cap O) = 75$. According to the principle of inclusion-exclusion, the number of students taking at least one juice is $n(A \cup O) = n(A) + n(O) - n(A \cap O) = 100 + 150 - 75 = 175$. The number of students taking neither juice is the complement of the union, $n(A' \cap O') = n((A \cup O)') = n(U) - n(A \cup O) = 400 - 175 = 225$. Option 2 is the number of students taking at least one juice. Option 3 and 4 represent common calculation errors such as neglecting the intersection or subtracting from a different base.

Using the Binomial Theorem, $4^n = (1+3)^n = 1 + 3n + \binom{n}{2}3^2 + \binom{n}{3}3^3 + \dots + 3^n$. Substituting this into the expression for set $A$, we get $4^n - 3n - 1 = 9[\binom{n}{2} + 3\binom{n}{3} + \dots + 3^{n-2}]$. Since the expression in the brackets is an integer for $n \ge 2$, every element of $A$ is a multiple of 9. For $n=1, A=0$; $n=2, A=9$; $n=3, A=54$. Set $B$ consists of all non-negative multiples of 9: $B = \{0, 9, 18, 27, 36, 45, 54, \dots\}$. Every element in $A$ exists in $B$, but elements like 18 and 27 are in $B$ but not in $A$. Thus, $A$ is a proper subset of $B$ ($A \subset B$). Option 2 is false as $B$ is larger. Option 3 is false because $A \neq B$. Option 4 is false because they share common elements like 0 and 9.