Search any question & find its solution
Question:
Answered & Verified by Expert
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
(i) If $x \in A$ and $A \in B$, then $x \in B$
(ii) If $A \subset B$ and $B \in C$, then $A \in C$
(iii) If $A \subset B$ and $B \subset C$, then $A \subset C$
(iv) If $A \not \subset B$ and $B \not \subset C$, then $A \not \subset C$
(v) If $x \in A$ and $A \not \subset B$, then $x \in B$
(vi) If $A \subset B$ and $x \notin B$, then $x \notin A$
(i) If $x \in A$ and $A \in B$, then $x \in B$
(ii) If $A \subset B$ and $B \in C$, then $A \in C$
(iii) If $A \subset B$ and $B \subset C$, then $A \subset C$
(iv) If $A \not \subset B$ and $B \not \subset C$, then $A \not \subset C$
(v) If $x \in A$ and $A \not \subset B$, then $x \in B$
(vi) If $A \subset B$ and $x \notin B$, then $x \notin A$
Solution:
2001 Upvotes
Verified Answer
(i) False. Let $A=\{1\}, B=\{\{1\}, 2\}$
$\therefore 1 \in \mathrm{A}$ and $A \in B$ but $1 \notin B$
So, $x \in \mathrm{A}$ and $A \in B$ need not imply that $x \in B$
(ii) False. Let $A=\{1\}, B=\{1,2\}$ and
$C=\{\{1,2\}, 3\}$
$\therefore A \subset B$ and $B \in C$ but $A \notin C$
Thus, $A \subset B$ and $B \in C$ need not imply that $A \in C$.
(iii) True. Let $x \in A$, Then
$A \subset B \Rightarrow x \in B$, then $B \subset C \Rightarrow x \in C$
Thus, $A \subset B$ and $B \subset C \Rightarrow A \subset C$
(iv) False. Let $A=\{1,2\}, B=\{2,3\}$ and $C=\{1,2,5\}$
Then $A \not \subset B$ and $B \not \subset C$ But $A \subset C$
Thus, $A \not \subset B$ and $B \not \subset C$ need not imply that $A \not \subset C$
(v) False. Let $A=\{1,2\}$ and $B=\{2,3,4,5\}$
Then $1 \in A$ and $A \not \subset B$ as $1 \notin B$
Thus, $x \in A$ and $A \not \subset B$ need not imply that $x \in B$
(vi) True. Let $A \subset B$, then
$x \in A \Rightarrow x \in B \Leftrightarrow x \notin B \Rightarrow x \notin A$
$\therefore 1 \in \mathrm{A}$ and $A \in B$ but $1 \notin B$
So, $x \in \mathrm{A}$ and $A \in B$ need not imply that $x \in B$
(ii) False. Let $A=\{1\}, B=\{1,2\}$ and
$C=\{\{1,2\}, 3\}$
$\therefore A \subset B$ and $B \in C$ but $A \notin C$
Thus, $A \subset B$ and $B \in C$ need not imply that $A \in C$.
(iii) True. Let $x \in A$, Then
$A \subset B \Rightarrow x \in B$, then $B \subset C \Rightarrow x \in C$
Thus, $A \subset B$ and $B \subset C \Rightarrow A \subset C$
(iv) False. Let $A=\{1,2\}, B=\{2,3\}$ and $C=\{1,2,5\}$
Then $A \not \subset B$ and $B \not \subset C$ But $A \subset C$
Thus, $A \not \subset B$ and $B \not \subset C$ need not imply that $A \not \subset C$
(v) False. Let $A=\{1,2\}$ and $B=\{2,3,4,5\}$
Then $1 \in A$ and $A \not \subset B$ as $1 \notin B$
Thus, $x \in A$ and $A \not \subset B$ need not imply that $x \in B$
(vi) True. Let $A \subset B$, then
$x \in A \Rightarrow x \in B \Leftrightarrow x \notin B \Rightarrow x \notin A$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.