Complement of a Set Questions in English

(A) The complement of a set $A$,denoted by $A^{\prime}$,consists of all elements in the universal set $U$ that are not in $A$.
By definition,$A$ and $A^{\prime}$ are disjoint sets,meaning they have no common elements.
Therefore,the intersection of a set and its complement is the empty set.
$A \cap A^{\prime} = \varnothing$

(A) We know that the complement of the universal set $U$ is the empty set,denoted by $\varnothing$.
So,$U^{\prime} = \varnothing$.
Therefore,$U^{\prime} \cap A = \varnothing \cap A = \varnothing$.
Thus,$U^{\prime} \cap A = \varnothing$.

(A) The probability of an event $A$ is given as $P(A) = \frac{2}{11}$.
The probability of the event 'not $A$',denoted as $P(\text{not } A)$ or $P(A')$,is calculated using the complement rule:
$P(\text{not } A) = 1 - P(A)$.
Substituting the given value:
$P(\text{not } A) = 1 - \frac{2}{11} = \frac{11 - 2}{11} = \frac{9}{11}$.

(A) Given that $P(A) = 0.42$,$P(B) = 0.48$,and $P(A \cap B) = 0.16$.
We need to find $P(\text{not } A)$,which is the complement of event $A$,denoted as $P(A^c)$ or $P(A')$.
The formula for the complement of an event is $P(A^c) = 1 - P(A)$.
Substituting the given value: $P(A^c) = 1 - 0.42 = 0.58$.

(A) Given that $P(B) = 0.48$.
The probability of the complement of an event $B$,denoted as $P(\text{not } B)$ or $P(B^c)$,is given by the formula:
$P(B^c) = 1 - P(B)$.
Substituting the given value:
$P(\text{not } B) = 1 - 0.48 = 0.52$.

(C) Using the Mean Value Theorem,$|\sin(\sqrt{n+1}) - \sin(\sqrt{n})| = |\cos(c) \cdot (\sqrt{n+1} - \sqrt{n})|$ for some $c \in (\sqrt{n}, \sqrt{n+1})$.
Since $\sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n+1} + \sqrt{n}}$,we have $|\sin(\sqrt{n+1}) - \sin(\sqrt{n})| = |\cos(c)| \cdot \frac{1}{\sqrt{n+1} + \sqrt{n}}$.
As $n \to \infty$,$\frac{1}{\sqrt{n+1} + \sqrt{n}} \to 0$.
Since $|\cos(c)| \le 1$,the expression $|\sin(\sqrt{n+1}) - \sin(\sqrt{n})|$ approaches $0$ as $n \to \infty$.
For any $\lambda > 0$,there exists an $N$ such that for all $n > N$,$|\sin(\sqrt{n+1}) - \sin(\sqrt{n})| < \lambda$.
Thus,$A_\lambda$ contains all natural numbers greater than $N$,making $A_\lambda$ an infinite set.
Consequently,the complement $A_\lambda^c$ contains only a finite number of elements (those $n \le N$ that do not satisfy the inequality).
Therefore,$A_{1/2}^c, A_{1/3}^c, A_{2/5}^c$ are all finite sets.

(C) The total number of subsets of the set $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ is $2^n$,where $n=9$.
Total subsets $= 2^9 = 512$.
We want to find the number of subsets that contain at least one odd number.
It is easier to calculate the complement: the number of subsets that contain $NO$ odd numbers.
$A$ subset contains no odd numbers if and only if all its elements are even.
The even numbers in the set are $\{2, 4, 6, 8\}$.
The number of subsets formed using only these even numbers is $2^4 = 16$.
These $16$ subsets include the empty set $\emptyset$.
Therefore,the number of subsets containing at least one odd number is Total subsets $-$ Subsets with only even numbers.
Required number $= 512 - 16 = 496$.