Sets

We are asked to find the number of elements in the set formed by the letters of the word "POSSESSIVENESS". The key point is that in a set, repeated elements are counted only once. Let's first list the distinct letters in "POSSESSIVENESS": The word "POSSESSIVENESS" consists of the following letters: The distinct letters are: Thus, the set formed by the distinct letters is: Therefore, the number of elements in the set is .

The correct answer is option (A):

We are given the expression , which represents the union of two sets:
refers to the elements in but not in ,
refers to the elements in but not in . The union of these two sets includes all elements in that are not in , plus all elements in that are not in . Now, the expression gives us all elements in either or , except the elements that are in both and (i.e., the intersection). Thus, the solution simplifies to:

The correct option is (D):

We are given the set , which means that for each element of , we have the condition: First, we solve for . Rearranging the inequality: Since is always positive for natural numbers, the inequality holds for all natural numbers. Next, we observe that the elements of must be natural numbers. We list the elements of that satisfy the condition , i.e., . The values of satisfying are: Thus, the elements of the set are . To find the mean value of the elements of , we calculate the average: Thus, the mean value of the elements of is .

The correct option is (D):

To find , we use the formula for the union of two sets: Substitute the given values:

The correct option is (C):

Step 1: Recall the definition of disjoint sets.

If , it means that sets and have no elements in common. Such sets are called disjoint sets.

Step 2: Analyze each option.

• (1) A and B are disjoint sets: This is true because directly implies that and are disjoint.
• (2) and : This is not necessarily true. For example, if and , then , but neither nor is empty.
• (3) and : This is also not necessarily true. For example, if and , then , but neither nor is empty.
• (4) All of these: This is incorrect because options (2) and (3) are not always true.

Final Answer: The correct statement is , which corresponds to option .

Correct answer:

Explanation:
Given:

This means that the intersection of sets A and B is equal to set B itself.
That implies every element of B is also present in A.

Therefore, B is a subset of A.

So, the correct relation is:

We are given that . Let's analyze this equation.
- The union of two sets, , contains all the elements that are in either or .
- The intersection of two sets, , contains only the elements that are in both and .

For the union and intersection of two sets to be equal, every element in must also be in , and vice versa. This implies that and must be the same set.

Thus, .

So, the correct option is (C):

Let represent the number of students who like music, and represent the number of students who like dance. From the given data:
Total students = 60
Students who like music
Students who like dance
Students who like neither = 5
Thus, the number of students who like either music or dance or both is: Now, applying the principle of inclusion and exclusion: Substitute and : Simplifying:

The correct option is (B):

The given sets are:
- (a) Null set
- (b) Singleton set
- (c) Infinite set

Now, we need to match the sets with the appropriate conditions.

(i)
(ii)
(iii)

Let's analyze each option:

- The Null set (a) has no elements, so it matches with (iii), where the condition is for even prime numbers, and the only even prime is 2, which is a singleton.
- The Singleton set (b) contains exactly one element, and it can be matched with (ii), where is a whole number and less than zero.
- The Infinite set (c) contains infinitely many elements, so it matches with (i), where is a real number.

So, the correct option is (D): (a)—(ii), (b}—(iii), (c)—(i)

We are asked to identify which of the following collections is not a set.

is a set because it is well-defined and includes all boys in a particular class.
is a set because it is a well-defined collection of numbers that follows a specific rule.
is because "most talented" is a subjective criterion, and it may differ from person to person. Therefore, this collection is not well-defined.
is incorrect, as the collection of ten most talented writers is not a set.

Thus, the collection of ten most talented writers of India is .

The correct option is (C): The collection of ten most talented writers of India.

We are given two sets:

and .

Step 1: Understanding the Sets

Both sets are defined using the same condition: all elements are multiples of 3, where is a natural number.

Since both sets are identical, we conclude that:

Step 2: Evaluating the Options

• Option 1: (True, but since , it is not the best choice)
• Option 2: (Same reasoning as above)
• Option 3: (Correct, since both sets are identical)
• Option 4: (Incorrect, because and contain only multiples of 3, not all natural numbers)

Final Answer: .

Let A = {1, 2, 3, 4, 5}.
(3) {3, 4} ∈ A
This statement means that the set {3, 4} is an element of set A. The elements of A are 1, 2, 3, 4, and 5. The set {3, 4} is not an element of A. So, {3, 4} ∈ A is incorrect.

(2) {3, 4} ⊆ A
This statement means that the set {3, 4} is a subset of A. A subset of A is a set containing only elements that are also in A. The set {3, 4} contains the elements 3 and 4, which are both elements of A. Thus, {3, 4} is a subset of A. So, {3, 4} ⊆ A is correct.

(1) {3, 4} ⊂ A
This statement means that the set {3, 4} is a proper subset of A. A proper subset of A is a subset of A that is not equal to A. Since {3, 4} is a subset of A and {3, 4} ≠ A, {3, 4} is a proper subset of A. So, {3, 4} ⊂ A is correct.

The incorrect statement is (3).

Final Answer: The final answer is 3