Question:
Suppose $ {{A}_{1}},{{A}_{2}}.....,{{A}_{30}} $ are thirty sets each with five elements and $ {{B}_{1}},{{B}_{2}}.....,{{B}_{n}} $ are 'n' sets each with three elements.
Let $ \underset{i=1}{\mathop{\overset{30}{\mathop{\cup }}\,}}\,\,\,{{A}_{i}}=\underset{j=1}{\mathop{\overset{n}{\mathop{\cup }}\,}}\,\,\,\,{{B}_{j}}=S.
$ Assume that each element of S belongs to exactly 10 of $ {{A}_{I}}'s $ and exactly 9 of $ {{B}_{j}}'s, $ then the value of $n$ is
Suppose $ {{A}_{1}},{{A}_{2}}.....,{{A}_{30}} $ are thirty sets each with five elements and $ {{B}_{1}},{{B}_{2}}.....,{{B}_{n}} $ are 'n' sets each with three elements.
Let $ \underset{i=1}{\mathop{\overset{30}{\mathop{\cup }}\,}}\,\,\,{{A}_{i}}=\underset{j=1}{\mathop{\overset{n}{\mathop{\cup }}\,}}\,\,\,\,{{B}_{j}}=S.
$ Assume that each element of S belongs to exactly 10 of $ {{A}_{I}}'s $ and exactly 9 of $ {{B}_{j}}'s, $ then the value of $n$ is
Updated On: Jun 23, 2024
- $ 90 $
- $ 15 $
- $ 9 $
- 45
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The Correct Option is D
Solution and Explanation
If elements are not repeated then number of elements in
$ {{A}_{1}}\cup {{A}_{2}}\cup {{A}_{3}}\cup ......\cup {{A}_{30}} $
is $ 30\times 5. $
but each element is used 10 time.
$ \therefore $ $ S=\frac{30\times 5}{10}=15 $ ...(i)
Similarly, if elements in $ {{B}_{1}},\,{{B}_{2}}.....{{B}_{n}} $
are not repeated, then total number of elements is 3n but each elements is repeated 9 times.
$ S=\frac{3n}{9} $
$ \Rightarrow $ $ 15=\frac{3n}{9}; $
[from E (i)]
$ \therefore $
$ n=45 $
$ {{A}_{1}}\cup {{A}_{2}}\cup {{A}_{3}}\cup ......\cup {{A}_{30}} $
is $ 30\times 5. $
but each element is used 10 time.
$ \therefore $ $ S=\frac{30\times 5}{10}=15 $ ...(i)
Similarly, if elements in $ {{B}_{1}},\,{{B}_{2}}.....{{B}_{n}} $
are not repeated, then total number of elements is 3n but each elements is repeated 9 times.
$ S=\frac{3n}{9} $
$ \Rightarrow $ $ 15=\frac{3n}{9}; $
[from E (i)]
$ \therefore $
$ n=45 $
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".