Understanding Mathematical Sets and Probabilities

The mathematical concepts of sets, especially when combined with probability theory, can be intricate and fascinating. In this article, we explore a specific problem involving sets and probabilities, and how we can solve it step-by-step. This problem is a typical example of how to approach and solve complex mathematical questions systematically.

Finding the Integer x: A Step-by-Step Guide

Given the sets M, T, and F, along with their respective probabilities, our goal is to determine the positive integer x. Here’s a detailed breakdown of the problem and its solution:

Step 1: Defining the Sample Space

The set M is defined as {1, 2, 3, ..., 20}. This means our sample space contains the first 20 positive integers. The probability of an element being in one of the subsets T and F is given relative to this sample space. Specifically, P(F) 0.25, P(T ∩ F) 0.1, and P(T ∪ F) 0.45.

Step 2: Applying Probability Rules

We can use the rule of probability to find the number of elements in each set. Since P(F) 0.25, we know that F contains 0.25 * 20 5 elements.

Step 3: Analyzing the Properties of Sets

The set T is defined as the divisors of 18. To find T, we first determine the divisors of 18, which are 1, 2, 3, 6, 9, and 18. This set has 6 elements. However, the problem states that F is not defined based on divisors of 18 but rather on a positive integer x with certain properties.

Step 4: Utilizing the Odd Number of Divisors

It is mentioned that the set F has an odd number of elements, which is 5. In number theory, only square numbers have an odd number of divisors. Therefore, we can infer that x must be a square number with exactly 5 divisors.

Step 5: Determining the Form of x

A number with exactly 5 divisors must be of the form ( k^4 ). This is because the number of divisors of a number ( n p_1^{e_1} p_2^{e_2} cdots p_k^{e_k} ) is given by ( (e_1 1)(e_2 1) cdots (e_k 1) ). For this product to be 5, the only possible combination is ( 5 5 times 1 ), which means x must be a fourth power of a prime number.

Step 6: Finding the Correct x

Given that ( x leq 20 ), we need to find the prime number ( k ) such that ( k^4 leq 20 ). Testing the prime numbers:

If ( k 2 ), then ( 2^4 16 leq 20 ). If ( k 3 ), then ( 3^4 81 ) which is greater than 20.

Hence, the only possible value for x is 16.

Conclusion

By understanding the properties of sets, the rules of probability, and the unique characteristics of square numbers, we were able to determine that the positive integer x is 16. Such problems combine algebra, number theory, and probability, making them excellent exercises for honing one’s mathematical skills.

This solution demonstrates a methodical approach to solving complex mathematical problems by breaking them down into manageable steps and utilizing fundamental principles from set theory and number theory. The importance of each property, such as the odd number of divisors, and the application of probability rules, are key elements in arriving at the correct solution.