Understanding the Algebraic System (M, *)

We are asked to analyze an algebraic system defined as (M, *). Here, M represents the set of all non-zero real numbers ($\mathbb{R} \setminus \{0\}$), and '*' is a binary operator defined by the rule $X*Y = \frac{XY}{4}$. The task is to identify which of the fundamental properties - Closure, Associative, Inverse, and Commutative - are satisfied by this system.

Verification of Properties

1. Closure Property

The closure property checks if, for any two elements $X$ and $Y$ chosen from the set M, the result of their operation, $X*Y$, remains within the set M.

• Let $X$ and $Y$ be any two elements from M. By definition of M, both $X$ and $Y$ are non-zero real numbers ($X \neq 0, Y \neq 0$).
• The operation is $X*Y = \frac{XY}{4}$.
• The product of two non-zero real numbers, $XY$, is itself a non-zero real number.
• Dividing this non-zero product $XY$ by 4 results in $\frac{XY}{4}$, which is also a non-zero real number.
• Therefore, the result $X*Y$ is always an element of M.
• Conclusion: The Closure Property holds true for the system (M, *).

2. Associative Property

The associative property is satisfied if, for any three elements $X, Y,$ and $Z$ in M, the grouping of operations does not change the outcome, meaning $(X*Y)*Z = X*(Y*Z)$.

• Let's evaluate the left side: $(X*Y)*Z$.
• First, $X*Y = \frac{XY}{4}$.
• Then, $\left(\frac{XY}{4}\right)*Z = \frac{(\frac{XY}{4})Z}{4} = \frac{XYZ}{16}$.
• Now, let's evaluate the right side: $X*(Y*Z)$.
• First, $Y*Z = \frac{YZ}{4}$.
• Then, $X*\left(\frac{YZ}{4}\right) = \frac{X(\frac{YZ}{4})}{4} = \frac{XYZ}{16}$.
• Since both sides yield the same result ($\frac{XYZ}{16}$), the order of operations does not matter.
• Conclusion: The Associative Property holds true for the system (M, *).

3. Inverse Property

For the inverse property to hold, two conditions must be met: there must be an identity element ($e$) in M such that $X*e = X$ for all $X \in M$, and for each $X \in M$, there must exist an inverse element ($X^{-1}$) in M such that $X*X^{-1} = e$.

• Identity Element ($e$): We seek an element $e \in M$ such that $X*e = X$ for all $X \in M$.
• Using the operator definition: $\frac{Xe}{4} = X$.
• Since $X$ is non-zero, we can divide by $X$: $\frac{e}{4} = 1$.
• Solving for $e$, we find $e = 4$. Since 4 is a non-zero real number, $e=4$ is in M.
• Inverse Element ($X^{-1}$): For each $X \in M$, we need an element $X^{-1} \in M$ such that $X*X^{-1} = e = 4$.
• Using the operator definition: $\frac{XX^{-1}}{4} = 4$.
• Multiplying both sides by 4 gives $XX^{-1} = 16$.
• Solving for $X^{-1}$, we find $X^{-1} = \frac{16}{X}$.
• For any non-zero real number $X$, the value $\frac{16}{X}$ is also a non-zero real number. Thus, the inverse $X^{-1}$ exists and is in M for every $X \in M$.
• Conclusion: The Inverse Property holds true for the system (M, *).

4. Commutative Property

The commutative property requires that the order of operands does not affect the result, meaning $X*Y = Y*X$ for all $X, Y \in M$.

• Calculate $X*Y$: $X*Y = \frac{XY}{4}$.
• Calculate $Y*X$: $Y*X = \frac{YX}{4}$.
• Since the multiplication of real numbers is commutative ($XY = YX$), it follows that $\frac{XY}{4}$ is equal to $\frac{YX}{4}$.
• Thus, $X*Y = Y*X$.
• Conclusion: The Commutative Property holds true for the system (M, *).

Summary of Properties Satisfied

The analysis shows that the algebraic system (M, *) satisfies the Closure Property (A), the Associative Property (B), the Inverse Property (C), and the Commutative Property (D).

Therefore, all the listed properties are satisfied by the given algebraic system.