JEE MAIN - Mathematics (2012 - No. 1)
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 :
35
25
53
52
Explanation
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 × 3 × 3 × 3 × 3 [$$ \because $$ no. of elements in set X = 5]
= 35
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 × 3 × 3 × 3 × 3 [$$ \because $$ no. of elements in set X = 5]
= 35
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