Composition Law Problem: Solving For Neutral Element & More

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Hey guys! Today, we're diving into a fascinating problem involving a composition law defined on real numbers. We'll be tackling the equation x * y = xy + x + y - 1 + 2^{xy}. Get ready to explore how to prove specific values, identify neutral elements, and solve for unknown numbers within this framework. Let's get started!

a) Proving 1 * 2 = 8

First, let's prove that 1 * 2 = 8 using the given composition law. This is a straightforward substitution, but it's crucial to follow the order of operations carefully. We'll replace x with 1 and y with 2 in the equation x * y = xy + x + y - 1 + 2^{xy}.

So, 1 * 2 = (1)(2) + 1 + 2 - 1 + 2^{(1)(2)}. Now, let's simplify this step-by-step:

  1. (1)(2) = 2
  2. 2 + 1 + 2 - 1 = 4
  3. 2^{(1)(2)} = 2^2 = 4

Therefore, 1 * 2 = 4 + 4 = 8. Ta-da! We've successfully shown that 1 * 2 indeed equals 8. This demonstrates how the composition law operates with specific numerical values. This first part is all about applying the formula directly. Understanding this basic application is fundamental to tackling the more complex parts of the problem. It's like building the foundation for a house – you need a solid base before you can start adding the walls and roof. Make sure you're comfortable with this substitution process before moving on! This foundational step helps to demystify the operation and allows you to see how the different components interact to produce a result. The key here is methodical substitution and simplification. It lays the groundwork for understanding the more abstract concepts that follow, like neutral elements and solving for unknowns.

b) Demonstrating e = 0 as the Neutral Element

Next up, we need to show that e = 0 is the neutral element of the composition law '*'. Remember, a neutral element (also called an identity element) is a value that, when combined with any other element using the given operation, leaves the other element unchanged. In mathematical terms, this means that for any real number x, x * 0 = x and 0 * x = x.

Let's test this with our composition law. We need to show that x * 0 = x and 0 * x = x. Let's start with x * 0:

x * 0 = x(0) + x + 0 - 1 + 2^{x(0)}

Simplifying this:

  1. x(0) = 0
  2. x + 0 - 1 = x - 1
  3. 2^{x(0)} = 2^0 = 1

So, x * 0 = 0 + x + 0 - 1 + 1 = x. Awesome! It checks out for x * 0. Now let's check 0 * x:

0 * x = 0(x) + 0 + x - 1 + 2^{0(x)}

Simplifying:

  1. 0(x) = 0
  2. 0 + x - 1 = x - 1
  3. 2^{0(x)} = 2^0 = 1

So, 0 * x = 0 + 0 + x - 1 + 1 = x. Double awesome! It also checks out for 0 * x. Since x * 0 = x and 0 * x = x for any real number x, we've successfully demonstrated that e = 0 is indeed the neutral element of the composition law '*'. This part delves into the heart of algebraic structures – the concept of a neutral element. The neutral element is like the 'zero' in addition or the 'one' in multiplication; it leaves other elements unchanged when combined. This property is crucial for defining many mathematical operations and structures. By proving that 0 is the neutral element, we're essentially establishing a fundamental characteristic of this particular composition law. The symmetry in this proof – showing both x * 0 = x and 0 * x = x – is also worth noting. In some operations, the order matters, but in this case, the composition law behaves consistently regardless of the order. This adds to our understanding of the law's properties and helps solidify the concept of the neutral element. It’s not just about plugging in numbers; it's about understanding the behavior of the operation and how it interacts with specific elements.

c) Determining the Non-Zero Natural Number n for which n * (1/n) = 0

Now, for the final part, we need to find the non-zero natural number n that satisfies the equation n * (1/n) = 0. This is where things get a little more interesting, and we'll need to combine our understanding of the composition law with some algebraic thinking.

Let's substitute x = n and y = 1/n into the composition law:

n * (1/n) = n(1/n) + n + (1/n) - 1 + 2^{n(1/n)}

Simplifying this equation:

  1. n(1/n) = 1
  2. 2^{n(1/n)} = 2^1 = 2

So, n * (1/n) = 1 + n + (1/n) - 1 + 2. Now we have:

n * (1/n) = n + (1/n) + 2

We're given that n * (1/n) = 0, so we can set up the equation:

0 = n + (1/n) + 2

To solve for n, let's get rid of the fraction by multiplying the entire equation by n:

0 = n^2 + 1 + 2n

Rearranging the terms, we get a quadratic equation:

n^2 + 2n + 1 = 0

This looks familiar, right? It's a perfect square trinomial! We can factor it as:

(n + 1)^2 = 0

Taking the square root of both sides:

n + 1 = 0

Solving for n:

n = -1

However, the question asks for a non-zero natural number. Natural numbers are positive integers (1, 2, 3, ...). Since -1 is not a natural number, there is no non-zero natural number n that satisfies the equation n * (1/n) = 0 for this composition law. This last part throws a curveball! It's not just about applying the formula and solving; it's about interpreting the solution within the context of the problem. We solved the quadratic equation perfectly, but the solution, n = -1, doesn't fit the criteria of being a non-zero natural number. This highlights the importance of always checking your answers against the original problem statement. Math isn't just about crunching numbers; it's about logical reasoning and understanding the constraints of the problem. The fact that there's no solution within the specified domain (non-zero natural numbers) is a valid and important finding. It shows us that not all equations have solutions, and sometimes the absence of a solution is the solution itself. It's a reminder that critical thinking and careful interpretation are just as crucial as the mathematical techniques we employ. This part is like the detective work of mathematics – piecing together the clues and making sure the final picture makes sense within the rules of the game.

Conclusion

So, there you have it! We've successfully navigated this composition law problem. We proved that 1 * 2 = 8, demonstrated that 0 is the neutral element, and determined that there's no non-zero natural number n for which n * (1/n) = 0. Awesome work, guys! These types of problems are great for building your understanding of mathematical operations and problem-solving skills. Remember, the key is to break down complex problems into smaller, manageable steps and to always double-check your answers within the given context. Keep practicing, and you'll become a math whiz in no time!