Let X=1,2,3,4,5.

Question

Let $X=\{1,2,3,4,5\}$. The number of different ordered pairs $(Y,Z)$ that can be formed such that $Y\subseteq X$, $Z\subseteq X$, and $Y\cap Z$ is empty, is:

Accepted Answer

For any element xi present in X, 4 cases arises while making subsets Y and Z.

Case- 1 : xi $$ \in $$ Y, xi $$ \in $$ Z $$ \Rightarrow $$ Y $$ \cap $$ Z $$ \ne $$ $$\phi $$

Case- 2 : xi $$ \in $$ Y, xi $$ \notin $$ Z $$ \Rightarrow $$ Y $$ \cap $$ Z = $$\phi $$

Case- 3 : xi $$ \notin $$ Y, xi $$ \in $$ Z $$ \Rightarrow $$ Y $$ \cap $$ Z = $$\phi $$

Case- 4 : xi $$ \notin $$ Y, xi $$ \notin $$ Z $$ \Rightarrow $$ Y $$ \cap $$ Z = $$\phi $$

$$ \therefore $$ For every element, number of ways = 3 for which Y $$ \cap $$ Z = $$\phi $$

$$ \Rightarrow $$ Total ways = 3 \times  3 \times  3 \times  3 \times  3 [$$ \because $$ no. of elements in set X = 5]

= 35

Solutions · 1