mathematics-class-11-unit-01-chapter-03-set-theory-l-3-3-4qtlloo9png

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-3 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Problem 1</B></Primary></p>
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<p>If $n(A)=6, n(B)=4$</p>
<p><span style="color:black">What is the minimum & maximum value of $n(A-B)$ ?</span></p>
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-3 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p>
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<p>$n(A)=n(A-B)+n(A \cap B) $</p>
<p>$\Rightarrow n(A-B)=n(A)-n(A \cap B) $</p>
<p>$= 6 - n(A \cap B)$</p>
<p>$\because A \cap B \subseteq B \Rightarrow n(A \cap B) \leq n(B) = 4$</p>
<p>$\therefore n(A-B) \geq 6-4=2 \leftarrow \text { Min. value }$</p>
<p>Also, if $A \cap B=\phi, n(A \cap B)=0$</p>
<p>$\therefore \operatorname{Max.} n(A-B)=6-0=6$</p>
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-3 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Problem 4</B></Primary></p>
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<p>Suppose 70% Indians like apple, and $82$ % like mango. Let $x$ % like both.</p>
<p>Find the min. & max. possible $x$.</p>
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-3 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p>
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<p>Given: $n(A)=70, \quad n(M)=82$</p>
<p>$n(A \cap M)=x$</p>
<p>Since $\quad n(A \cup M) \leq 100,$</p>
<p>$n(A)+n(M)-n(A \cap M) \leq 100 $</p>
<p>$\Rightarrow 70+82-x \leq 100 $
$\Rightarrow x \geq 52$</p>
<p>Also, $A \cap M \subseteq A$</p>
<p>$\therefore n(A \cap M) \leq n(A)=70\qquad\quad$
So, $52 \leq x \leq 70$</p>
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<p>[<strong>Recall</strong> : $\quad P(A)=$ all subsets of $A$.</p>
<p>$n(P(A))=2^{n(A)}$]</p>
<p>Since $n(\phi)=0, \quad n(P(\phi))=2^{0}=1$.</p>
<p>$\Rightarrow n(P(P(\phi)))=2^1=2 $</p>
<p>$\Rightarrow n(P(P(P(\phi))))=2^2=4 .$</p>
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-3 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Problem 6</B></Primary></p>
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<p>If $n(A)=m, \quad n(B)=n$,</p>
<p>$n(P(A))-n(P(B))=112$.</p>
<p>Find m and n.</p>
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-3 </B></Success></p>
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<p>Since, $2^m-2^n=112(\Rightarrow m>n) $</p>
<p>$\Rightarrow 2^n\left(2^{m-n}-1\right)=112=16 \times 7=2^4 \times 7 $</p>
<p>$\Rightarrow n=4$ & $2^{m-n}-1=7$</p>
<p>$\Rightarrow n=4$ & $2^{m-n}=8=2^3 $</p>
<p>$\Rightarrow n=4$ & $m-n=3 $</p>
<p>$\Rightarrow n=4$ & $m=7 $</p>
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<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Problem 7</B></Primary></p>
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<p>$A=\{1,2,3,4\}, B=\{2,4,6\}$</p>
<p>Find the no. of sets $C$ such that</p>
<p>$A \cap B \subseteq C \subseteq A \cup B . $</p>
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-3 </B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p>
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<p>$A \cap B=\{2,4\}$</p>
<p>$A \cup B=\{1,2,3,4,6\}$</p>
<p>So, {$2,4$} $\subseteq C$ $\subseteq${$1,2,3,4,6$}
$\Rightarrow \quad C$={2,4} $\cup C^{\prime}$</p>
<p>$\text { where }C^{\prime} \subseteq${1,3,6}</p>
<p>$\therefore$ Number of such C= Number of $C^{\prime}$ $\subseteq${1,3,6}</p>
<p>$=n(P(${$1,3,6$}$)) $
$=2^3=8 .$</p>
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<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Problem 9</B></Primary></p>
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<p>$A=\{n^3+(n+1)^3+(n+2)^3: n \in \mathbb{N}\}$</p>
<p>$B=\{9 n: n \in \mathbb{N}\}$</p>
<p>Then which of the following is/are true?</p>
<p>(a) $A \subseteq B$</p>
<p>(b) $B \subseteq A$</p>
<p>(c) $A=B$</p>
<p>(d) $A \subsetneq B$.</p>
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<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Set theory  lecture-3 </B></Success></p>
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<p>$A=\{36,2^3+3^3+4^3, \cdots\}$</p>
<p>Smallest no. in $A$ is 36 .</p>
<p>So, $A \neq B$ and $B \nsubseteq A$.</p>
<p>(b) & (c) are FALSE.</p>
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