Question

Let x= {1, 2, 3, 4, 5} The number of different ordered pairs (y, z) that can be formed such ordered pairs (y , z) that can be formed such that $y⊑x,z⊑x$ and $y\cap z$ is empty is

• A. $2^5$
• B. $5^3$
• C. $5^2$
• D. $3^5$

Answer 1

The correct option is D $3^5$

Given,

$x=\{1,2,3,4,5\}$

$y\subseteq x,z\subseteq x.$

It means all element of $z$ are in $x$

and all elements of $y$ are in $x$.

and $y\cap z=0$

Now, set take a subset $b$

such that,

case $I,b\in Y$ $b\in Z$

case $IIb\in yb\notin z$

case $IIIb\notin 45\in Z$

case $IV$ b $\notin y$ b $\notin z$

According to ques. Case I is not possible.

Considering Case II, III and IV.

$\text{Total ways}(4\cap 2=0)=3\times 3\times 3\times 3\times 3$

$=3^5$

Option (D) is correct.