GATE Overflow Test Series | Discrete Mathematics | Test 1 | Question: 7

Question

Consider the following two statements about sets $X ,Y$ and $Z$
$S_1:$ If $X \subseteq Y$ and $Y \subseteq Z,$ then $X \subseteq Z.$
$S_2:$ If $X \in Y$ and $Y \in Z,$ then $X \in Z$

• A. Only $S_1$ is true
• B. Only $S_2$ is true
• C. Both $S_1$ and $S_2$ are true
• D. Neither $S_1$ nor $S_2$ is true

Answer 1

For $S_1$ if we draw a Venn diagram we can see that $X$ will be inside $Z$ and so $S_1$ is TRUE.

For $S_2,$ consider the sets $X = \{1\}, Y = \{\{1\}\}, Z = \{\{\{1\}\}\}.$ Here, $X \in Y, Y \in Z,$ but $X \notin Z.$ So, $S_2$ is FALSE

Answer 2

Let $,X=\{1\},\;\;Y=\{1,2\},\;\; Z=\{1,2,3\}$

$S_{1}:\text{If}\;X\subseteq Y,Y\subseteq Z, \text{then} \:X\subseteq Z$

We can write like this way, $[(X\subseteq Y)\wedge (Y\subseteq Z)] \rightarrow \:(X\subseteq Z)$

Let $X = \{\ a\},\;\;Y=\{\{\ a\}\},Z = \{\{\{\ a\}\}\}$

We can easily see, $X \in Y,\;\;Y\in Z,\,\;X \notin Z$

So, the correct answer is $(A).$