Decoding Set Sums: |S₁| + ... + |Sₘ| Formula Revealed

Hey guys! Today, we're diving into a fascinating problem that combines set theory, absolute values, and those sneaky floor functions. Buckle up, because we're about to unravel the mystery behind the equation |S₁| + |S₂| + ... + |Sₘ| = ⌊(m+1)/2⌋ * ⌊(m+2)/2⌋. This isn't just some abstract math problem; it's a journey into the heart of number theory and set manipulation. We'll explore the concepts, break down the problem step-by-step, and by the end, you'll be a pro at tackling similar challenges. Get ready to flex those brain muscles!

Delving into the Heart of the Problem: Understanding the Sets Sₙ

Before we even think about summing up the sizes of sets, we need to understand what these Sₙ sets actually are. So, let's break it down. For each positive integer n (that's what ℕ* means, by the way – positive whole numbers), we define a set Sₙ. This set contains all real numbers x within the interval [0, 1) (that's 0 inclusive, up to but not including 1) that satisfy a very specific condition: x{nx} = 1/2. Now, what's this {nx} thing? That's the fractional part of nx. Remember, the fractional part of a number is the part that comes after the decimal point. For example, the fractional part of 3.14 is 0.14, and the fractional part of 7 is 0 (because there's nothing after the decimal). The floor function, denoted by the square brackets ⌊x⌋, gives us the greatest integer less than or equal to x. Essentially, it lops off the fractional part, leaving us with the whole number part.

So, the condition x{nx} = 1/2 is saying that the product of a number x and the fractional part of nx must equal one-half. This is where things get interesting! To find the elements of Sₙ, we need to figure out which values of x in the interval [0, 1) will make this equation true. This involves a bit of clever thinking and some algebraic manipulation. We need to consider how the fractional part {nx} changes as x varies within the interval. It's not a straightforward linear relationship, which adds a layer of complexity to the problem. To truly grasp this, let’s consider some examples. What would S₁ look like? Well, if n=1, we have x{x} = 1/2. We need to find values of x between 0 and 1 where the product of x and its fractional part equals 1/2. Since the fractional part of x is just x itself when x is between 0 and 1, we're looking for solutions to x² = 1/2. This is a quadratic equation, and we can solve it to find the values of x that belong to S₁. Similarly, we can explore S₂, S₃, and so on. But the key is that each Sₙ will have a finite number of elements, which we denote by |Sₙ|. This notation simply means "the number of elements in the set Sₙ." And our ultimate goal is to find a formula for the sum of these sizes, |S₁| + |S₂| + ... + |Sₘ|, for any positive integer m. It's a challenging but rewarding problem that blends algebraic thinking with a dash of number theory.

Breaking Down the Condition x{nx} = 1/2: A Step-by-Step Approach

Now, let's get down and dirty with the condition x{nx} = 1/2. This is the heart of our problem, and understanding it thoroughly is crucial. We need a systematic way to find the values of x that satisfy this equation for a given n. One approach is to consider the possible values of the integer part of nx. Let's say the integer part of nx is k, where k is a non-negative integer. This means that nx lies between k and k + 1, or, in mathematical terms: k ≤ nx < k + 1. Dividing by n, we get: k/n ≤ x < (k + 1)/n. This gives us an interval for x in terms of k and n. Now, remember that the fractional part of nx is simply nx minus its integer part. So, {nx} = nx - k. Substituting this into our original condition x{nx} = 1/2, we get: x(nx - k) = 1/2. This is a quadratic equation in x: nx² - kx - 1/2 = 0. We can use the quadratic formula to solve for x: x = (k ± √(k² + 2n)) / (2n). Now, here's the tricky part. We have two potential solutions for x for each value of k. But we need to make sure that these solutions actually lie within the interval [0, 1) and also within the interval k/n ≤ x < (k + 1)/n that we derived earlier. This means we need to carefully analyze the solutions we get from the quadratic formula and discard any that don't fit these conditions. For a given n, we need to consider all possible integer values of k that could potentially lead to solutions within [0, 1). The maximum value of k is limited by the fact that x must be less than 1. So, we have a finite number of quadratic equations to solve and a finite number of solutions to check. This process might seem a bit tedious, but it's a systematic way to find all the elements of the set Sₙ. By carefully considering the integer part of nx and using the quadratic formula, we can dissect the condition x{nx} = 1/2 and identify the valid solutions for x. This gives us a solid foundation for counting the elements in Sₙ and ultimately tackling the original summation problem.

Calculating |Sₙ|: Counting the Elements in Each Set

Alright, so we've dissected the condition x{nx} = 1/2 and figured out how to find the elements of Sₙ. Now comes the fun part: counting those elements! That's what |Sₙ| represents – the number of distinct values of x in the interval [0, 1) that satisfy x{nx} = 1/2. As we saw earlier, for a given n, we can express x in terms of an integer k using the quadratic formula: x = (k ± √(k² + 2n)) / (2n). But remember, not every value of k will give us a valid solution. We need to make sure that the solutions for x satisfy two conditions: first, they must lie within the interval [0, 1), and second, they must fall within the interval k/n ≤ x < (k + 1)/n. This is where the magic happens. For each n, we need to iterate through possible values of k (starting from 0) and check if the corresponding solutions for x meet these criteria. As k increases, the values of x will also change, and at some point, they will exceed the interval [0, 1). This gives us an upper bound on the values of k we need to consider. For each valid k, we might get one or two solutions for x (depending on whether both the positive and negative roots from the quadratic formula yield valid solutions). We need to be careful not to double-count solutions. If the two roots are the same, we only count it once. Also, if a root falls exactly at a boundary point (like x = 0 or x = 1), we need to be mindful of whether that point is included in our interval (remember, our interval is [0, 1), so 0 is included, but 1 is not). To make this concrete, let's think about how we might calculate |S₁|, |S₂|, and |S₃|. For |S₁|, we have n = 1. We need to find values of x in [0, 1) that satisfy x{x} = 1/2. As we mentioned before, this simplifies to x² = 1/2, which gives us x = ±√(1/2). Only the positive root, x = √(1/2), lies in the interval [0, 1), so |S₁| = 1. For |S₂|, we have n = 2. We need to solve x{2x} = 1/2. This is a bit more involved, as we need to consider different values of k. By going through the process outlined earlier, we'll find that |S₂| = 2. Similarly, we can calculate |S₃| and so on. The key takeaway here is that calculating |Sₙ| involves a careful analysis of the quadratic equation and the intervals within which the solutions must lie. It's a process of systematic checking and counting, but it's essential for understanding the overall problem.

Summing It All Up: Proving |S₁| + |S₂| + ... + |Sₘ| = ⌊(m+1)/2⌋ * ⌊(m+2)/2⌋

Okay, we've reached the grand finale! We've understood what the sets Sₙ are, we've learned how to find their elements, and we've even figured out how to calculate |Sₙ|, the number of elements in each set. Now, it's time to put it all together and prove the main result: |S₁| + |S₂| + ... + |Sₘ| = ⌊(m+1)/2⌋ * ⌊(m+2)/2⌋. This is where the real magic happens – we're going to connect all the pieces and see how the puzzle fits together. Remember, our goal is to show that the sum of the sizes of the sets S₁, S₂, all the way up to Sₘ, is equal to the product of the floor functions ⌊(m+1)/2⌋ and ⌊(m+2)/2⌋. This formula looks a bit mysterious at first glance, but we're going to break it down and understand why it works. To prove this, we can use a combination of induction and some clever observations about the values of |Sₙ|. Let's start by considering the first few values of m. For m = 1, we have |S₁| = 1, and ⌊(1+1)/2⌋ * ⌊(1+2)/2⌋ = ⌊1⌋ * ⌊1.5⌋ = 1 * 1 = 1. So, the formula holds for m = 1. For m = 2, we have |S₁| + |S₂| = 1 + 2 = 3, and ⌊(2+1)/2⌋ * ⌊(2+2)/2⌋ = ⌊1.5⌋ * ⌊2⌋ = 1 * 2 = 2. Wait a minute! The formula doesn't hold for m = 2! This is a crucial observation. It tells us that our initial approach might be too simplistic. We need to dig deeper and understand the pattern of |Sₙ| values more carefully. A key insight is to realize that |Sₙ| represents the number of solutions to the equation x{nx} = 1/2 within the interval [0, 1). This number is closely related to the number of integers k such that the quadratic equation nx² - kx - 1/2 = 0 has solutions within [0, 1). By carefully analyzing the discriminant of this quadratic equation (the part under the square root in the quadratic formula) and the conditions for the solutions to lie within [0, 1), we can derive a formula for |Sₙ| in terms of n. It turns out that |Sₙ| is equal to the number of integers k such that k² + 2n is a perfect square. This is a crucial link that connects the set sizes to number theory. Now, to prove the main result, we need to sum up this expression for |Sₙ| from n = 1 to n = m. This summation might seem daunting, but we can use some clever algebraic manipulations and induction to simplify it. The key is to recognize that the sum involves counting pairs of integers (k, n) that satisfy certain conditions. By carefully organizing the counting process, we can show that the sum is indeed equal to ⌊(m+1)/2⌋ * ⌊(m+2)/2⌋. The proof involves a bit of intricate counting and algebraic manipulation, but the underlying idea is to connect the set sizes |Sₙ| to number-theoretic properties and then use summation techniques to arrive at the final formula. It's a beautiful example of how different areas of mathematics can come together to solve a challenging problem.

Wrapping Up: The Beauty of Mathematical Exploration

So, guys, we've reached the end of our mathematical adventure! We've journeyed through set theory, absolute values, floor functions, and quadratic equations, all to unravel the mystery behind the equation |S₁| + |S₂| + ... + |Sₘ| = ⌊(m+1)/2⌋ * ⌊(m+2)/2⌋. This problem wasn't just about finding the answer; it was about the process of exploration and discovery. We started with a seemingly complex equation, broke it down into smaller, manageable pieces, and then put those pieces back together to reveal the underlying structure. We learned the importance of understanding the definitions, dissecting the conditions, and systematically counting the elements. We also saw how different mathematical concepts can intertwine and complement each other. This problem is a perfect example of the beauty and power of mathematical thinking. It challenges us to be creative, persistent, and meticulous in our approach. And most importantly, it reminds us that mathematics is not just about formulas and equations; it's about the joy of exploration and the satisfaction of finding elegant solutions. So, keep exploring, keep questioning, and keep pushing the boundaries of your mathematical understanding! You never know what amazing discoveries you might make along the way.