Question:
If $A$ and $B$ are finite sets and , ${A \subset B}$ then
If $A$ and $B$ are finite sets and , ${A \subset B}$ then
Updated On: Apr 8, 2024
- $n (A \cup B)\, =\, n(A)$
- $n (A \cap B)\, = \,n(B)$
- $n (A \cup B)\, =\, n(B)$
- $n (A \cap B) \,=\, \phi$
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The Correct Option is C
Solution and Explanation
We have, $A \subset B$
$\therefore A \cap B=A \Rightarrow n(A \cap B)=n(A) \dots$(i)
Again, we know that
$n(A \cup B)=n(A)+n(B)-n(A \cap B)$
$\Rightarrow n(A \cup B)=n(A)+n(B)-n(A) $ [from E (i)]
$\Rightarrow n(A \cup B)=n(B)$
$\therefore A \cap B=A \Rightarrow n(A \cap B)=n(A) \dots$(i)
Again, we know that
$n(A \cup B)=n(A)+n(B)-n(A \cap B)$
$\Rightarrow n(A \cup B)=n(A)+n(B)-n(A) $ [from E (i)]
$\Rightarrow n(A \cup B)=n(B)$
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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 "|".