Differentiation (Part-38) | S N De

 $\,1.\,$ Define with examples:

$\,(i)\,$ Finite and infinite sets $\,(ii)\,$ Null set  

$\,(iii)\,$ Universal set  $\,(iv)\,$ Singleton set

$\,(v)\,$ Equal sets $\,(vi)\,$ Subset and proper subset  $\,(vii)\,$ Union of two sets

$\,(viii)\,$ Intersection of two sets  $\,(ix)\,$ Disjoint sets

$\,(x)\,$ Complement of a set  $\,(xi)\,$ Difference of two sets $\,(xii)\,$ Power Set

$\,2.\,$ Distinguish

$\,(i)\,$ Null set and universal set

$\,(ii)\,$ Union and intersection of two sets

$\,(iii)\,$ Subset and proper subset

$\,(iv)\,$ Union and difference of two sets

$\,(v)\,$ Universal set and complement of a set

$\,3.\,$ Prove that every set is a subset of its own. 

$\,4.\,$ Show that null set is a subset of all sets.

$\,5.\,$ Write short notes on:

$\,(i)\,$ Power set $\,(ii)\,$ Venn diagram

$\,(iii)\,$ Duality

Sol. From  question no. $\,(1)\,$ to $\,(5)\,$ follow S.N. De text book.  

$\,6.\,$  If $\,A=\{a, b, c\}$, name $\,(i)\,$  the subsets of $\,A,\, \, (ii)\,$ the proper subsets of A.

Sol. (i) The subsets of $\,A,\, \, \text{are}\quad \phi, \{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\,\,\text{or}\,\,A. $

(ii) The proper subsets of $\,A,\, \, \text{are}\quad \phi, \{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}. $

$\,7.\,$ Define power set of a set $\,A.\,$ Find the power set of $\,A=\{\{1\}, \{2, 3\}\}$ .

Sol. We know that power set of a given set $\,A\,$ is the set of all its subsets and is denoted by $\,P(A)\,.$ Symbolically, $\,\,P(A)=\{X | X \subseteq A\}.$ 

The power set of $\,A=\{\phi,\{1\}, \{2, 3\}, \{1,2,3\}\}.$

$8\,.$ If  $\,A=\{1, 2, 3, 4\},\, B=\{2, 4, 5, 8\}\,$ and $\,C=\{3, 4, 5, 6, 7\},$ 

find $\,(i)\, A \cup B\,\,(ii)\, B \cap C \\ (iii)\, A \cup (B \cup C)\,\,(iv)\, A \cup (B \cap C) $

Sol. $(i)\,\, A \cup B\\=\{1, 2, 3, 4\} \cup \{2, 4, 5, 8\}\\=\{1,2,3,4,5,8\} \\ (ii) B \cap C\\=\{2, 4, 5, 8\} \cap \{3, 4, 5, 6, 7\}\\=\{4,5\} \cdots (1) \\ (iii)\,\, \text{We first compute},\,\, \\ B \cup C=\{2, 4, 5, 8\} \cup\{3, 4, 5, 6, 7\}\\~~~~~~~~~~~=\{2,3,4,5,6,7,8\} \\ A \cup (B \cup C)\\=\{1, 2, 3, 4\} \cup \{2,3,4,5,6,7,8\}\\=\{1,2,3,4,5,6,7,8\} \\ (iv)\,\, A \cup (B \cap C)\\=\{1, 2, 3, 4\} \cup \{4,5\}\,\,[\text{By (1)}]\\=\{1,2,3,4,5\}$

$\,9.\,$ If $\,\,P=\{a,b,c,d,e\},\,\,Q=\{a,e,i,o,u\},\,\,$ prove that $\,(i)\, P \subset P \cup Q \,\, (ii)\,\, P \cap Q \subset P$

Sol. $\,(i)\,P \cup Q\\=\{a,b,c,d,e\} \cup \{a,e,i,o,u\}\\=\{a,b,c,d,e,i,o,u\} \\ \therefore P \subset P\cup Q \\ (ii)\,\, P\cap Q=\{a,e\} \\ \therefore P\cap Q \subset P$

$\,10.\,$ If $\,A \subseteq B\,\,$ and $\,\,B \subseteq C,\,\,$ prove that, $\,\,A \subseteq C$

Sol. Since $\,A \subseteq B,\,\, x\in A \Rightarrow x \in B \cdots(1)$

Again,$\,\,B \subseteq C,\,\,x \in B \Rightarrow x\in C \cdots(2)$

From (1) and (2), it follows $\,x \in A \Rightarrow x \in C \\ \text{so},\,\,A \subseteq C.$ 

$\,11.\,$ If $\,\,A \cup B=B,\,$  show that, $\,A \subseteq B.$

Sol. Let $\,\,x \in A .$ 

Now ,$\,\,x \in A \\ \Rightarrow x \in A \cup B \\ \Rightarrow x \in B\,\,[\text{Since,}\,\,A \cup B=B]\\ x \in A \Rightarrow x \in B \\ \therefore A \subseteq B.$ 

$\,12.\,$ If $\,A \subseteq B,\,\,$ prove that , $\,\,A-B=\phi.$

Sol. Since $\,\,A \subseteq B,\,\,\forall x \in A \Rightarrow x \in B.$

So, there does not have any element of $\,A\,$, which does not belong to $\,B.$ 

Hence, $\,\,A-B=\phi.$

$\,13(i).\,$ Applying set operations prove that, $\,3+4=7.$

Sol. Let $\,A=\{a,b,c\},\,\,B=\{p,q,r,t\} \\ \therefore n(A)=3,\,\,n(B)=4 \\ \text{Again,}\,\, A \cap B=\phi,\,\,n(A \cap B)=0  \rightarrow (1)\,\\ A \cup B=\{a,b,c,p,q,r,t\} \\ \therefore\,\, n(A \cup B)=7  \\ \Rightarrow n(A)+n(B)=7\,\,[\text{By (1)}] \\ \Rightarrow 3+4=7$

$\,13(ii).\,$ Represent the set in Roster form:

$\,A = \{(x, y) : (x, y) \,\,\text{is the coordinate}\\ \text{ of point of intersection of line}\,\, y = x \\ \text{and curve}\,\, y = e^x\}$ .

Sol. Since $\,\,y=x\,\,$ and $\,\,y=e^x\,\,$ do not have a common solution, so $\,A =\phi.$

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