If $X=\{1,2,3, \ldots, 10\}$ and $A=\{1,2,3,4,5\}$.
Then, the number of subsets $B$ of $X$ such that $A-B=\{4\}$ is
- $2^5$
- $2^4$
- $2^5-1$
- 1
The Correct Option is A
Solution and Explanation
The correct option is(A): 25.
Given, \(X=\{1,2,3, \ldots, 10\}\)
and \(A=\{1,2,3,4,5\}\)
Now, \(A-B=\{4\}\)
\(\Rightarrow \, B=A-\{4\}=\{1,2,3,4,5\}-\{4\}\)
\(=\{1,2,3,5\}\)
\(\therefore \, X-\{4\}-B\)
\(=\{1,2,3, \ldots, 10\}-\{4\}-\{1,2,3,5\}\)
\(=\{6,7,8,9,10\}\)
\(\therefore\) Number of subsets \(B\) of \(X\) i.e., \(\{6,7,8,9,10\}\)
\(=2^{5}\).
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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 "|".