Sets

We are given a set with 9 elements, and we need to find the number of subsets of the set that contain at most 4 elements.

The total number of subsets of a set with elements is given by:

For a set of 9 elements, the total number of subsets is:

Now, we need to calculate the number of subsets that contain at most 4 elements. This means we need to count subsets containing 0, 1, 2, 3, and 4 elements. We use the combination formula to find the number of subsets of each size:

• Number of subsets with 0 elements:
• Number of subsets with 1 element:
• Number of subsets with 2 elements:
• Number of subsets with 3 elements:
• Number of subsets with 4 elements:

Now, we sum these values to find the total number of subsets containing at most 4 elements:

The number of subsets of the set that contain at most 4 elements is 256.

The correct answer is P only.

Step-by-step Explanation:

The set X is defined as {1, π, {42, √2}, {1, 3}}. We will now evaluate each of the given statements (P, Q, R) one by one:

• Statement P: π ∈ X
• Statement Q: {1, 3} ⊆ X
• Statement R: {1, 7} ⊆ X

Conclusion:

Only the statement P (π ∈ X) is true, so the correct answer is P only.

We are given the set , which contains 9 elements. We are asked to find how many subsets can be formed that contain exactly 4 elements.
The number of ways to select 4 elements from a set of 9 elements is given by the combination formula: where (the total number of elements in the set) and (the number of elements to be chosen).
Thus, the number of subsets containing exactly 4 elements is:

The correct option is (A) :

The set is defined as , which translates to .
The set is defined as , which simplifies to .
The intersection of sets and , denoted , includes the elements that are common to both sets.

The correct option is (A) :

We are given the following information: - - - We need to find . We know the formula for the union of two sets: We can also express as: Substitute this into the union formula: Simplifying: So:

The correct option is (C) :

Given: Let .

We analyze the polynomial modulo 3:

Using Fermat's Little Theorem ( ):

This shows:

Therefore, the set contains no natural numbers:

Since , we know that is always non-negative, and so for all values of .
Therefore, the function maps to values in the range .

Step 1: Understand the Problem We are given two finite sets and , each containing 3 elements. We need to determine the total number of possible mappings (functions) from to .
Step 2: Recall the Formula for Number of Mappings If has elements and has elements, the total number of mappings (functions) from to is given by: This is because each element of has choices in , and the choices are independent.
Step 3: Apply the Formula Here, and each have 3 elements. Therefore:
- (number of elements in )
- (number of elements in )
The total number of mappings from to is:
Step 4: Verify the Answer The total number of mappings is , which corresponds to option (E).
Final Answer: The total number of mappings from to is: