Question

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

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

Answer 1

The correct option is B

$3^5$

Explanation for the correct answer:

Find the number of different ordered pairs $\left(y,z\right)$ :

Given, $y\subseteq x,z\subseteq x$ and $x=\left\{1,2,3,4,5\right\}$

Let, $\mathrm{a}\in x$, then

There will be four cases

Case 1: $\mathrm{a}\in \mathrm{y},\mathrm{a}\in \mathrm{z}$

Case 2: $\mathrm{a}\in \mathrm{y},\mathrm{a}\notin \mathrm{z}$

Case 3: $\mathrm{a}\notin \mathrm{y},\mathrm{a}\in \mathrm{z}$

case 4: $\mathrm{a}\notin \mathrm{y},\mathrm{a}\notin \mathrm{z}$

As, we require $y\cap z=\phi$

Thus cases $1,2$ and $3$ satisfy $y\cap z=\phi$.

Therefore, $y\cap z=\phi$ has $3$ options.

Thus, the total possibilities for $5$ elements of $x=3^5$

Hence, the correct option is B.