Question:
For any two sets $A$ and $B$, $A-(A-B)$ equals
For any two sets $A$ and $B$, $A-(A-B)$ equals
Updated On: Apr 27, 2024
- $B$
- $A-B$
- $A \cap B$
- $A^{c} \cap B^{c}$
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The Correct Option is C
Solution and Explanation
Now, $A-\left(A-B\right)=A-\left(A \cap B^{c}\right)$
$=A \cap\left(A\cap B^{c}\right)^{c}$
$=A\cap\left(A^{c}\cup B\right)$
$=\left(A\cap A^{c}\right)\cup\left(A \cap B\right)$
$=A\cap B$
$=A \cap\left(A\cap B^{c}\right)^{c}$
$=A\cap\left(A^{c}\cup B\right)$
$=\left(A\cap A^{c}\right)\cup\left(A \cap B\right)$
$=A\cap B$
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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 "|".