Let Z Be A Set, Investigate The Properties Of The Law Defined By P * Q = P + Q + Pq. 1) Show That * Is A Commutative And Associative Internal Composition Law. 2) Show That * Has A Neutral Element. 3) What Are The Invertible Elements?

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Introduction

In this article, we delve into the fascinating world of abstract algebra by examining a specific binary operation defined on the set of integers, denoted by Z. This exploration allows us to understand the fundamental properties that govern mathematical structures. We will investigate the characteristics of the operation, including its commutativity, associativity, the existence of an identity element, and the identification of invertible elements.

Let's define our binary operation, denoted by *, as follows: for any two integers p and q, the operation p * q is defined as:

p * q = p + q + pq

Our primary goal is to meticulously analyze this operation and uncover its underlying algebraic properties. We will begin by demonstrating that * is indeed a law of internal composition, meaning that the result of the operation on any two integers is also an integer. Subsequently, we will explore the key properties of commutativity and associativity, which dictate the order in which elements can be combined without affecting the result. Furthermore, we will investigate the existence of an identity element, a special element that leaves other elements unchanged when combined using the operation. Finally, we will delve into the concept of invertible elements, which possess an inverse within the set such that their combination yields the identity element. Through this comprehensive analysis, we will gain a deeper understanding of the algebraic structure defined by the set of integers under the operation *.

1. Proving Internal Composition, Commutativity, and Associativity

Internal Composition

First, let's establish that * is a law of internal composition on Z. This means that for any two integers p and q, the result of p * q must also be an integer. Since p and q are integers, their sum (p + q) is also an integer. Similarly, the product of p and q (pq) is an integer. Finally, the sum of three integers (p + q + pq) is also an integer. Therefore, p * q = p + q + pq is an integer, confirming that * is indeed a law of internal composition on Z.

Commutativity

Next, we will demonstrate that the operation * is commutative. An operation is commutative if the order of the operands does not affect the result. In other words, we need to show that p * q = q * p for all integers p and q. Let's start by expanding both sides of the equation:

p * q = p + q + pq
q * p = q + p + qp

Since addition and multiplication are commutative operations on integers, we can rewrite the second equation as:

q * p = p + q + pq

Comparing the expressions for p * q and q * p, we observe that they are identical. Therefore, p * q = q * p for all integers p and q, proving that the operation * is commutative.

Associativity

Now, let's examine whether the operation * is associative. An operation is associative if the grouping of operands does not affect the result. In other words, we need to show that (p * q) * r = p * (q * r) for all integers p, q, and r. Let's expand both sides of the equation step by step:

(p * q) * r = (p + q + pq) * r
            = (p + q + pq) + r + (p + q + pq)r
            = p + q + pq + r + pr + qr + pqr

Now, let's expand the right-hand side:

p * (q * r) = p * (q + r + qr)
            = p + (q + r + qr) + p(q + r + qr)
            = p + q + r + qr + pq + pr + pqr

Comparing the expressions for (p * q) * r and p * (q * r), we can see that they are identical. Therefore, (p * q) * r = p * (q * r) for all integers p, q, and r, proving that the operation * is associative. This associativity property is a cornerstone of many algebraic structures, allowing us to manipulate expressions involving the * operation with confidence.

2. Identifying the Identity Element

An identity element (often called a neutral element) is a special element within a set that, when combined with any other element using a given operation, leaves the other element unchanged. In the context of our operation *, we seek an integer e such that for any integer p:

p * e = p
e * p = p

Since we have already established that the operation * is commutative, we only need to verify one of these equations. Let's use the first equation, p * e = p, and substitute the definition of the operation:

p + e + pe = p

Now, we need to solve this equation for e. Subtracting p from both sides, we get:

e + pe = 0

We can factor out e from the left side:

e(1 + p) = 0

This equation holds true if either e = 0 or 1 + p = 0. However, 1 + p = 0 only holds for p = -1, while we are looking for an identity element that works for all integers p. Therefore, the only solution that satisfies the equation for all integers p is e = 0. Let's verify this by substituting e = 0 back into the original equation:

p * 0 = p + 0 + p(0) = p

This confirms that 0 is indeed the identity element for the operation *. This identity element plays a crucial role in understanding the structure of the set of integers under the operation *, as it provides a reference point for other elements.

3. Determining Invertible Elements

An invertible element is an element that, when combined with another element (its inverse) using the given operation, results in the identity element. In our case, we seek to find integers p that have an inverse p' such that:

p * p' = 0
p' * p = 0

Again, since the operation * is commutative, we only need to consider one of these equations. Let's use the first equation, p * p' = 0, and substitute the definition of the operation:

p + p' + pp' = 0

Now, we need to solve this equation for p'. We can rearrange the equation to isolate p':

p' + pp' = -p
p'(1 + p) = -p

If 1 + p is not equal to 0 (i.e., p ≠ -1), we can divide both sides by 1 + p to obtain:

p' = -p / (1 + p)

For p' to be an integer, -p / (1 + p) must be an integer. This means that (1 + p) must divide -p. We can rewrite -p as -(1 + p) + 1. Therefore, the condition becomes:

(1 + p) divides -(1 + p) + 1

Since (1 + p) always divides -(1 + p), the condition simplifies to:

(1 + p) divides 1

The only integers that divide 1 are -1 and 1. Therefore, we have two possibilities:

  1. 1 + p = 1, which implies p = 0. In this case, p' = -0 / (1 + 0) = 0, so the inverse of 0 is 0.
  2. 1 + p = -1, which implies p = -2. In this case, p' = -(-2) / (1 + (-2)) = 2 / (-1) = -2, so the inverse of -2 is -2.

Now, let's consider the case where p = -1. From the equation p'(1 + p) = -p, we get p'(1 + (-1)) = -(-1), which simplifies to p'(0) = 1. This equation has no solution for p', meaning that -1 does not have an inverse under the operation *.

Therefore, the only invertible elements in Z under the operation * are 0 and -2. The inverse of 0 is 0, and the inverse of -2 is -2. This limited set of invertible elements highlights a key characteristic of this particular binary operation. The concept of invertible elements is crucial for understanding the algebraic structure, as it determines which elements can be