Identity Element And Inverse Under The Commutative Operation * On Real Numbers
In the fascinating realm of mathematics, operations on sets form the bedrock of algebraic structures. Among these operations, the concept of a commutative operation holds significant importance. An operation is commutative if the order in which the elements are combined does not affect the result. In simpler terms, for a commutative operation denoted by *, x * y = y * x for all elements x and y in the set.
This article delves into the intricacies of a specific commutative operation defined on the set of real numbers (R). We will explore the operation * defined by x * y = x + y + 3xy, where x and y are real numbers. Our primary objectives are to determine the identity element e of R under this operation and to find the inverse of an element x belonging to R. This exploration will not only enhance our understanding of commutative operations but also provide a solid foundation for tackling more complex algebraic problems.
Unveiling the Commutative Nature of the Operation *
Before embarking on our quest to find the identity element and inverse, it is crucial to establish the commutative property of the operation *. This property ensures that the order in which we combine elements does not alter the outcome. To demonstrate commutativity, we need to show that x * y = y * x for all real numbers x and y.
Let's consider two real numbers, x and y. According to the definition of the operation *:
x * y = x + y + 3xy
Now, let's reverse the order and compute y * x:
y * x = y + x + 3yx
Observing both expressions, we can see that x + y is the same as y + x, and 3xy is the same as 3yx (due to the commutative property of multiplication of real numbers). Therefore,
x * y = y * x
This equality confirms that the operation * is indeed commutative on the set of real numbers. This foundational understanding paves the way for our next step: finding the identity element.
The Quest for the Identity Element e
The identity element plays a pivotal role in algebraic structures. In the context of an operation, the identity element is a special element that, when combined with any other element using the operation, leaves the other element unchanged. In other words, for an operation * and a set S, an element e in S is the identity element if x * e = x and e * x = x for all x in S.
Our mission now is to find this elusive identity element e for the operation * defined on the set of real numbers. To achieve this, we will employ the definition of the identity element and solve for e. Let e be the identity element. Then, for any real number x, we must have:
x * e = x
Substituting the definition of the operation *:
x + e + 3xe = x
Our goal is to isolate e and determine its value. To do so, we can subtract x from both sides of the equation:
e + 3xe = 0
Now, we can factor out e from the left-hand side:
e(1 + 3x) = 0
This equation holds true if either e = 0 or (1 + 3x) = 0. However, the latter condition would imply that x = -1/3, which would restrict the identity property to only one specific value of x. Since the identity element must hold for all real numbers x, we conclude that the only possibility is:
e = 0
Therefore, the identity element for the operation * on the set of real numbers is 0. This means that for any real number x, x * 0 = x and 0 * x = x. We have successfully discovered the identity element, a crucial component in understanding the behavior of this operation.
Unveiling the Inverse Element
Having found the identity element, our next objective is to determine the inverse of an element x in the set of real numbers under the operation *. The inverse of an element is another element that, when combined with the original element using the operation, results in the identity element.
Formally, for an element x in a set S with an operation * and an identity element e, the inverse of x, denoted as x⁻¹, is an element in S such that:
x * x⁻¹ = e
and
x⁻¹ * x = e
In our case, we want to find the inverse of a real number x under the operation *, where the identity element e is 0. Let x⁻¹ represent the inverse of x. Then, according to the definition of the inverse:
x * x⁻¹ = 0
Substituting the definition of the operation *:
x + x⁻¹ + 3xx⁻¹ = 0
Our aim is to solve this equation for x⁻¹, which will give us the inverse of x. To do this, we can rearrange the equation:
x⁻¹(1 + 3x) = -x
Now, we can isolate x⁻¹ by dividing both sides of the equation by (1 + 3x), provided that (1 + 3x) is not equal to zero:
x⁻¹ = -x / (1 + 3x)
This equation gives us the inverse of x under the operation *, except for the case when the denominator (1 + 3x) is equal to zero. Let's examine this case:
1 + 3x = 0
Solving for x:
x = -1/3
This reveals that the inverse of x exists for all real numbers except x = -1/3. When x = -1/3, the denominator becomes zero, and the inverse is undefined. Therefore, the inverse of a real number x under the operation * is given by:
x⁻¹ = -x / (1 + 3x), for x ≠ -1/3
In summary, we have successfully determined the inverse of an element x in the set of real numbers under the operation *, with the exception of x = -1/3. This completes our exploration of the identity element and inverse for this specific commutative operation.
Conclusion: A Journey Through Commutative Operations
In this article, we embarked on a journey to unravel the properties of a commutative operation defined on the set of real numbers. We began by establishing the commutative nature of the operation * defined by x * y = x + y + 3xy. This foundation allowed us to delve deeper into the concepts of the identity element and the inverse of an element.
We successfully identified the identity element e as 0, demonstrating its unique property of leaving any element unchanged when combined under the operation *. Furthermore, we determined the inverse of an element x as x⁻¹ = -x / (1 + 3x), with the exception of x = -1/3. This exploration has not only provided us with specific results for this operation but has also illuminated the general principles of commutative operations, identity elements, and inverses.
The knowledge gained from this exploration serves as a valuable stepping stone for further investigations into algebraic structures and their properties. Understanding commutative operations and their associated elements is crucial for tackling more advanced mathematical concepts and problem-solving in various fields of science and engineering. As we continue our mathematical journey, the insights gained here will undoubtedly prove to be invaluable.
