Identity Element And Inverse In Commutative Operation On Real Numbers

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In the fascinating world of abstract algebra, we often encounter operations defined on sets, and these operations can exhibit various properties. One such property is commutativity, which simplifies many calculations and allows us to explore the structure of the set more deeply. In this article, we delve into a specific operation, denoted by , defined on the set R of real numbers. This operation is given by the formula $x^ y = x + y + 3xy$. Our primary focus will be to investigate this operation's commutative nature and, more importantly, to determine the identity element e of R under this operation and the inverse of an element x in R.

Verifying the Commutative Property

Before we embark on the quest for the identity element and the inverse, let's first confirm that the operation * is indeed commutative. Recall that an operation is commutative if the order of the elements does not affect the result. In other words, for any two real numbers x and y, we must have $x^* y = y^* x$. Let's verify this using the definition of our operation:

xβˆ—y=x+y+3xyx^* y = x + y + 3xy

yβˆ—x=y+x+3yxy^* x = y + x + 3yx

Since addition and multiplication are commutative in the set of real numbers, we have x + y = y + x and 3xy = 3yx. Therefore,

xβˆ—y=x+y+3xy=y+x+3yx=yβˆ—xx^* y = x + y + 3xy = y + x + 3yx = y^* x

This confirms that the operation * is commutative, a crucial property that simplifies our subsequent calculations.

Finding the Identity Element e

Now that we've established commutativity, let's turn our attention to finding the identity element e in R under the operation *. An identity element is a special element that, when combined with any other element using the operation, leaves the other element unchanged. Formally, e is the identity element if for any x in R,

xβˆ—e=x=eβˆ—xx^* e = x = e^* x

Since we know the operation is commutative, we only need to verify one of these equalities, say $x^* e = x$. Using the definition of the operation, we can write this as:

x+e+3xe=xx + e + 3xe = x

To find e, we need to solve this equation. Let's subtract x from both sides:

e+3xe=0e + 3xe = 0

Now, we can factor out e:

e(1+3x)=0e(1 + 3x) = 0

This equation must hold for all real numbers x. If we choose x = -1/3, the term (1 + 3x) becomes zero, and the equation holds regardless of the value of e. However, for any other value of x, the term (1 + 3x) is non-zero, and the only way for the equation to hold is if e = 0. Therefore, the identity element under the operation * is:

e=0e = 0

We can verify this by plugging e = 0 back into the definition of the operation:

xβˆ—0=x+0+3x(0)=xx^* 0 = x + 0 + 3x(0) = x

0βˆ—x=0+x+3(0)x=x0^* x = 0 + x + 3(0)x = x

This confirms that 0 is indeed the identity element for the operation *.

Determining the Inverse of an Element x

With the identity element in hand, we can now proceed to find the inverse of an element x in R under the operation *. The inverse of an element x, denoted by x', is another element in R such that when combined with x using the operation, the result is the identity element. In other words,

xβˆ—xβ€²=e=xβ€²βˆ—xx^* x' = e = x'^* x

Since we know the operation is commutative and that e = 0, we can write this as:

xβˆ—xβ€²=x+xβ€²+3xxβ€²=0x^* x' = x + x' + 3xx' = 0

Our goal is to find x' in terms of x. Let's rearrange the equation:

xβ€²+3xxβ€²=βˆ’xx' + 3xx' = -x

Factor out x':

xβ€²(1+3x)=βˆ’xx'(1 + 3x) = -x

Now, we can solve for x', provided that (1 + 3x) is not zero:

xβ€²=βˆ’x/(1+3x)x' = -x / (1 + 3x)

This formula gives us the inverse x' of an element x under the operation *, except for the case when the denominator (1 + 3x) is zero. This occurs when:

1+3x=01 + 3x = 0

3x=βˆ’13x = -1

x=βˆ’1/3x = -1/3

So, the inverse of x exists for all real numbers except x = -1/3. When x = -1/3, the element does not have an inverse under this operation. In summary, the inverse of x is:

x' = -x / (1 + 3x)$, for $x β‰  -1/3

Conclusion

In this exploration of the operation * defined on the set of real numbers by $x^* y = x + y + 3xy$, we have successfully demonstrated its commutative nature, found the identity element e = 0, and determined the inverse x' of an element x to be $x' = -x / (1 + 3x)$, except when x = -1/3. This exercise highlights the importance of understanding algebraic structures and their properties, such as commutativity, identity elements, and inverses, which are fundamental concepts in abstract algebra and have wide-ranging applications in mathematics and other fields. Further investigations could explore other properties of this operation, such as associativity, and its relationships with other algebraic structures.

Exploring the Identity Element and Inverse in the Commutative Operation

In the realm of mathematics, operations on sets play a crucial role in defining structures and relationships. When an operation exhibits the property of commutativity, meaning the order of elements does not affect the result, it opens doors to simplified calculations and deeper insights. This article focuses on a particular commutative operation, denoted by , defined on the set R of real numbers. The operation is given by the formula $x^ y = x + y + 3xy$. Our primary objective is to determine the identity element, denoted by e, of R under this operation and to find the inverse of an element x in R. This exploration will enhance our understanding of algebraic structures and their properties.

Understanding Commutativity

Before diving into the specifics of the identity element and inverse, it is essential to establish the commutative nature of the operation *. An operation is commutative if for any two elements x and y in the set, the following holds true:

xβˆ—y=yβˆ—xx^* y = y^* x

In the context of our operation *, this translates to:

x+y+3xy=y+x+3yxx + y + 3xy = y + x + 3yx

Since addition and multiplication are commutative operations within the set of real numbers, it is evident that the equation holds. Therefore, the operation * is indeed commutative. This property significantly simplifies our subsequent calculations, as we can freely interchange the order of elements without affecting the outcome.

Unveiling the Identity Element

With the commutative nature of the operation * confirmed, our next endeavor is to identify the identity element e. The identity element is a unique element within the set that, when combined with any other element using the operation, leaves the latter unchanged. Formally, e is the identity element if for any x in R:

xβˆ—e=x=eβˆ—xx^* e = x = e^* x

Given the commutativity of the operation, we only need to verify one of the equalities, say $x^* e = x$. Substituting the definition of the operation, we get:

x+e+3xe=xx + e + 3xe = x

To solve for e, we can subtract x from both sides of the equation:

e+3xe=0e + 3xe = 0

Factoring out e gives us:

e(1+3x)=0e(1 + 3x) = 0

This equation must hold true for all real numbers x. Consider the case when x = -1/3. In this scenario, the term (1 + 3x) becomes zero, and the equation holds irrespective of the value of e. However, for any other value of x, the term (1 + 3x) is non-zero. Therefore, the only way for the equation to hold is if e = 0. Thus, the identity element under the operation * is:

e=0e = 0

To verify this, we can substitute e = 0 back into the definition of the operation:

xβˆ—0=x+0+3x(0)=xx^* 0 = x + 0 + 3x(0) = x

0βˆ—x=0+x+3(0)x=x0^* x = 0 + x + 3(0)x = x

This confirms that 0 is indeed the identity element for the operation *.

Finding the Inverse of an Element

Now that we have identified the identity element, we can proceed to determine the inverse of an element x in R under the operation *. The inverse of an element x, denoted by x', is another element in R that, when combined with x using the operation, yields the identity element. Mathematically, this is expressed as:

xβˆ—xβ€²=e=xβ€²βˆ—xx^* x' = e = x'^* x

Since the operation is commutative and e = 0, we can write:

xβˆ—xβ€²=x+xβ€²+3xxβ€²=0x^* x' = x + x' + 3xx' = 0

Our objective is to express x' in terms of x. Let's rearrange the equation:

xβ€²+3xxβ€²=βˆ’xx' + 3xx' = -x

Factoring out x', we get:

xβ€²(1+3x)=βˆ’xx'(1 + 3x) = -x

Solving for x', provided that (1 + 3x) is not zero, we obtain:

xβ€²=βˆ’x/(1+3x)x' = -x / (1 + 3x)

This formula provides the inverse x' of an element x under the operation *, except for the case when the denominator (1 + 3x) is zero. This occurs when:

1+3x=01 + 3x = 0

3x=βˆ’13x = -1

x=βˆ’1/3x = -1/3

Therefore, the inverse of x exists for all real numbers except x = -1/3. When x = -1/3, the element does not have an inverse under this operation. In summary, the inverse of x is:

x' = -x / (1 + 3x)$, for $x β‰  -1/3

Synthesis

In this exploration of the operation * defined on the set of real numbers by $x^* y = x + y + 3xy$, we have successfully established its commutative nature, identified the identity element e = 0, and derived the inverse x' of an element x as $x' = -x / (1 + 3x)$, with the exception of x = -1/3. This exercise underscores the significance of understanding algebraic structures and their properties, including commutativity, identity elements, and inverses, which are fundamental concepts in abstract algebra with widespread applications in mathematics and other disciplines. Further investigations could delve into other properties of this operation, such as associativity, and its relationships with other algebraic structures.

Exploring Identity Element and Inverse Under the Operation * on Real Numbers

In the realm of abstract algebra, operations defined on sets form the bedrock of various mathematical structures. These operations, depending on their properties, can lead to fascinating insights and applications. One such property is commutativity, where the order of elements in an operation does not influence the outcome. This article delves into a specific commutative operation, denoted by , defined on the set R of real numbers, expressed as $x^ y = x + y + 3xy$. Our primary focus will be on identifying the identity element e within R under this operation and determining the inverse of an element x belonging to R. This exploration is crucial for understanding the algebraic nature of this operation and its implications.

Demonstrating Commutativity

Before embarking on the quest for the identity element and the inverse, it is imperative to establish that the operation * indeed adheres to the commutative property. As a reminder, an operation is commutative if, for any two elements x and y in the set, the following equation holds true:

xβˆ—y=yβˆ—xx^* y = y^* x

Applying this to our operation *, we need to verify if:

x+y+3xy=y+x+3yxx + y + 3xy = y + x + 3yx

Given the commutative nature of addition and multiplication within the set of real numbers, it is clear that the equation holds. Consequently, the operation * is commutative. This property streamlines our subsequent calculations, as we can freely interchange the order of elements without altering the result.

Identifying the Identity Element

Having confirmed the commutative nature of the operation *, our next objective is to pinpoint the identity element e. The identity element is a unique member of the set that, when combined with any other element using the operation, leaves the original element unchanged. Formally, e is the identity element if, for any x in R:

xβˆ—e=x=eβˆ—xx^* e = x = e^* x

Due to the commutativity of the operation, we only need to verify one of these equalities, for instance, $x^* e = x$. Substituting the definition of the operation, we obtain:

x+e+3xe=xx + e + 3xe = x

To isolate e, we can subtract x from both sides of the equation:

e+3xe=0e + 3xe = 0

Factoring out e yields:

e(1+3x)=0e(1 + 3x) = 0

This equation must be satisfied for all real numbers x. If we consider the case where x = -1/3, the term (1 + 3x) becomes zero, and the equation holds regardless of the value of e. However, for any other value of x, the term (1 + 3x) is non-zero. Therefore, the only way for the equation to hold is if e = 0. Consequently, the identity element under the operation * is:

e=0e = 0

To validate this, we can substitute e = 0 back into the definition of the operation:

xβˆ—0=x+0+3x(0)=xx^* 0 = x + 0 + 3x(0) = x

0βˆ—x=0+x+3(0)x=x0^* x = 0 + x + 3(0)x = x

This confirms that 0 is indeed the identity element for the operation *.

Determining the Inverse of an Element

With the identity element identified, we can now proceed to determine the inverse of an element x in R under the operation *. The inverse of an element x, denoted by x', is another element in R that, when combined with x using the operation, results in the identity element. Mathematically, this is expressed as:

xβˆ—xβ€²=e=xβ€²βˆ—xx^* x' = e = x'^* x

Given the commutativity of the operation and the fact that e = 0, we can write:

xβˆ—xβ€²=x+xβ€²+3xxβ€²=0x^* x' = x + x' + 3xx' = 0

Our objective is to express x' in terms of x. Rearranging the equation, we get:

xβ€²+3xxβ€²=βˆ’xx' + 3xx' = -x

Factoring out x', we obtain:

xβ€²(1+3x)=βˆ’xx'(1 + 3x) = -x

Solving for x', provided that (1 + 3x) is not zero, we arrive at:

xβ€²=βˆ’x/(1+3x)x' = -x / (1 + 3x)

This formula provides the inverse x' of an element x under the operation *, except for the case when the denominator (1 + 3x) is zero. This occurs when:

1+3x=01 + 3x = 0

3x=βˆ’13x = -1

x=βˆ’1/3x = -1/3

Thus, the inverse of x exists for all real numbers except x = -1/3. When x = -1/3, the element does not possess an inverse under this operation. In summary, the inverse of x is:

x' = -x / (1 + 3x)$, for $x β‰  -1/3

Conclusion and Insights

In this exploration of the operation * defined on the set of real numbers by $x^* y = x + y + 3xy$, we have successfully demonstrated its commutative nature, identified the identity element e = 0, and derived the inverse x' of an element x as $x' = -x / (1 + 3x)$, with the exception of x = -1/3. This exercise underscores the significance of understanding algebraic structures and their properties, including commutativity, identity elements, and inverses, which are fundamental concepts in abstract algebra with wide-ranging applications in mathematics and other disciplines. Further investigations could delve into other properties of this operation, such as associativity, and its relationships with other algebraic structures. This exploration highlights the interconnectedness of mathematical concepts and the importance of rigorous analysis in uncovering the underlying structure of mathematical systems.