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If A = P {1, 2} where P denotes the power set, then which one of the following is correct?
- a){1,2} ⊂ A
- b)1 ε A
- c)φ ∉ A
- d){1,2} ε A
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
If A = P {1, 2} where P denotes the power set, then which one of the f...
A = P {1 ,2} = {φ, {1}, {2}, {1 ,2 }}
From above, it is clear that {1 , 2} ε A
From above, it is clear that {1 , 2} ε A
Most Upvoted Answer
If A = P {1, 2} where P denotes the power set, then which one of the f...
Understanding the Power Set
The power set P{1, 2} includes all possible subsets of the set {1, 2}. Let's first enumerate the elements of this power set:
- The subsets of {1, 2} are:
- The empty set: {}
- The set containing just the first element: {1}
- The set containing just the second element: {2}
- The set containing both elements: {1, 2}
Thus, the power set P{1, 2} is:
P{1, 2} = { {}, {1}, {2}, {1, 2} }
This is the set A.
Analyzing the Options
Now, let's evaluate each of the provided options:
- Option a: {1, 2} ⊆ A
- This is incorrect because {1, 2} is not a subset of A; rather, it is an element of A.
- Option b: 1 ∈ A
- This is also incorrect. The number 1 is not an element of A; the elements of A are sets, not individual numbers.
- Option c: ∅ ∉ A
- This statement is incorrect as well. The empty set ∅ is indeed an element of A.
- Option d: {1, 2} ∈ A
- This is correct. The set {1, 2} is one of the elements in the power set P{1, 2}.
Conclusion
Thus, the correct answer is option 'D': {1, 2} ∈ A. The confusion often arises when distinguishing between elements and subsets within the context of power sets.
The power set P{1, 2} includes all possible subsets of the set {1, 2}. Let's first enumerate the elements of this power set:
- The subsets of {1, 2} are:
- The empty set: {}
- The set containing just the first element: {1}
- The set containing just the second element: {2}
- The set containing both elements: {1, 2}
Thus, the power set P{1, 2} is:
P{1, 2} = { {}, {1}, {2}, {1, 2} }
This is the set A.
Analyzing the Options
Now, let's evaluate each of the provided options:
- Option a: {1, 2} ⊆ A
- This is incorrect because {1, 2} is not a subset of A; rather, it is an element of A.
- Option b: 1 ∈ A
- This is also incorrect. The number 1 is not an element of A; the elements of A are sets, not individual numbers.
- Option c: ∅ ∉ A
- This statement is incorrect as well. The empty set ∅ is indeed an element of A.
- Option d: {1, 2} ∈ A
- This is correct. The set {1, 2} is one of the elements in the power set P{1, 2}.
Conclusion
Thus, the correct answer is option 'D': {1, 2} ∈ A. The confusion often arises when distinguishing between elements and subsets within the context of power sets.
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Community Answer
If A = P {1, 2} where P denotes the power set, then which one of the f...
A = P {1 ,2} = {φ, {1}, {2}, {1 ,2 }}
From above, it is clear that {1 , 2} ε A
From above, it is clear that {1 , 2} ε A
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