Answer
Verified
358.3k+ views
Hint: We are given a set say S as {0, 1, 2}. We will first find the power set of S. The power set of any set contains all the subsets of the given set. So, we will find the subsets of S = {0, 1, 2}. Then we will make a set using a subset of S which will make our power set. At last, the cardinality of the power set is the total element of the power set, so that we count the elements and get our solution.
Complete step-by-step answer:
We are given a set S as {0, 1, 2}. We are asked to find the cardinality of the power set of S. We will first look for the power set of S and then find further things. Now, we know that the power set of any set is the collection of all possible subsets of the given set S.
Now, as we have S as {0, 1, 2}. So the possible subset of S are
\[\phi ,\left\{ 0 \right\},\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 0,1 \right\},\left\{ 1,2 \right\},\left\{ 0,2 \right\},\left\{ 0,1,2 \right\}\]
So, these are our subsets.
Now, the power set is a collection of all the subsets as elements. So, the power set will become
\[P\left( S \right)=\left\{ \left\{ 0 \right\},\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 0,1 \right\},\left\{ 1,2 \right\},\left\{ 0,2 \right\},\left\{ 0,1,2 \right\},\phi \right\}\]
Now, we look for the cardinality of the power set. The cardinality of the set is the total number of elements contained in that set. Our power set contains 8 elements, so we get that cardinality of the power set of S = {0, 1, 2} as 8.
So, the correct answer is “Option A”.
Note: This can be done in an alternate method. We have the set as {0, 1, 2}. We have to find the cardinality of the power set. We know that the power set is the collection of all the subsets of the given set and the total number of elements of the power set is given as
\[\text{Number of elements of power set}={{2}^{n}}\]
where n is the total elements in the given set.
Our set {0, 1, 2} has three elements. So, n = 3 implies that the number of elements in the power set is \[{{2}^{3}}\] that is 8.
The cardinality of the power set is the number of elements in the power set. From the above, we have the power set as 8 elements. Therefore, the cardinality of the power set of {1, 2, 0} is 8.
Complete step-by-step answer:
We are given a set S as {0, 1, 2}. We are asked to find the cardinality of the power set of S. We will first look for the power set of S and then find further things. Now, we know that the power set of any set is the collection of all possible subsets of the given set S.
Now, as we have S as {0, 1, 2}. So the possible subset of S are
\[\phi ,\left\{ 0 \right\},\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 0,1 \right\},\left\{ 1,2 \right\},\left\{ 0,2 \right\},\left\{ 0,1,2 \right\}\]
So, these are our subsets.
Now, the power set is a collection of all the subsets as elements. So, the power set will become
\[P\left( S \right)=\left\{ \left\{ 0 \right\},\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 0,1 \right\},\left\{ 1,2 \right\},\left\{ 0,2 \right\},\left\{ 0,1,2 \right\},\phi \right\}\]
Now, we look for the cardinality of the power set. The cardinality of the set is the total number of elements contained in that set. Our power set contains 8 elements, so we get that cardinality of the power set of S = {0, 1, 2} as 8.
So, the correct answer is “Option A”.
Note: This can be done in an alternate method. We have the set as {0, 1, 2}. We have to find the cardinality of the power set. We know that the power set is the collection of all the subsets of the given set and the total number of elements of the power set is given as
\[\text{Number of elements of power set}={{2}^{n}}\]
where n is the total elements in the given set.
Our set {0, 1, 2} has three elements. So, n = 3 implies that the number of elements in the power set is \[{{2}^{3}}\] that is 8.
The cardinality of the power set is the number of elements in the power set. From the above, we have the power set as 8 elements. Therefore, the cardinality of the power set of {1, 2, 0} is 8.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Why Are Noble Gases NonReactive class 11 chemistry CBSE
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
At which age domestication of animals started A Neolithic class 11 social science CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE