Understanding the Binary Operation on Integer Numbers

The question asks us to determine the algebraic structure of the set of integer numbers, denoted by G, under a specific operation defined as \(a.b = a - b\) for all \(a, b \in G\). We need to examine the properties of this Binary Operation to see which of the given options best describes the structure \( (G, .) \).

Checking if it's a Binary Operation (Closure Property)

A operation '.' on a set G is called a Binary Operation if for every pair of elements \(a, b \in G\), the result \(a.b\) is also an element of G. This property is called closure.

In this case, the set is G, which is the set of all integer numbers (\( \mathbb{Z} \)), and the operation is subtraction (\(a.b = a - b\)).

Let \(a\) and \(b\) be any two integer numbers. Their difference, \(a - b\), is also an integer number.

For example:

• If \(a = 5\) and \(b = 3\), then \(a - b = 5 - 3 = 2\). Since 5, 3, and 2 are all integers, the operation is closed for this pair.
• If \(a = -2\) and \(b = 7\), then \(a - b = -2 - 7 = -9\). Since -2, 7, and -9 are all integers, the operation is closed for this pair.
• If \(a = 0\) and \(b = -4\), then \(a - b = 0 - (-4) = 4\). Since 0, -4, and 4 are all integers, the operation is closed for this pair.

Since the difference of any two integer numbers is always an integer number, the set G is closed under the operation \(a.b = a - b\). Therefore, the operation \(a.b = a - b\) is indeed a Binary Operation on the set of integer numbers G.

Investigating Group Properties for the Algebraic Structure

For the set G with the operation \(a.b = a - b\) to be a group, it must satisfy four properties: closure, associativity, existence of an identity element, and existence of an inverse element for each element. We already know it satisfies closure (it's a Binary Operation). Let's check the others.

Checking Associativity

An operation '.' on a set G is associative if for all \(a, b, c \in G\), \((a.b).c = a.(b.c)\).

For the operation \(a.b = a - b\):

\((a.b).c = (a - b).c = (a - b) - c = a - b - c\)

\(a.(b.c) = a.(b - c) = a - (b - c) = a - b + c\)

For the operation to be associative, we need \(a - b - c = a - b + c\) for all integers a, b, c. This is only true if \(c = -c\), which means \(c = 0\). Since this must hold for *all* integer numbers c, and it does not (e.g., if c=5, \(5 \neq -5\)), the operation is not associative.

For example, let \(a=10, b=4, c=2\):

\((10.4).2 = (10 - 4).2 = 6.2 = 6 - 2 = 4\)

\(10.(4.2) = 10.(4 - 2) = 10.2 = 10 - 2 = 8\)

Since \(4 \neq 8\), \((10.4).2 \neq 10.(4.2)\). The operation is not associative.

Checking for Identity Element

An element \(e \in G\) is an identity element if for all \(a \in G\), \(a.e = a\) and \(e.a = a\).

For the operation \(a.b = a - b\):

From \(a.e = a\), we have \(a - e = a\), which implies \(e = 0\).

From \(e.a = a\), we have \(e - a = a\), which implies \(e = a + a = 2a\).

For an identity element to exist, the value of \(e\) must be the same for all elements \(a \in G\). We found \(e=0\) from the first equation and \(e=2a\) from the second. These two expressions for \(e\) are not equal for all integers \(a\) (they are equal only when \(a = 0\)). Therefore, there is no identity element in G under this operation.

Since the operation is not associative and there is no identity element, the set of integer numbers G under the operation \(a.b = a - b\) does not form a group.

Considering Other Options: Quasi-group

A quasi-group is a set G with a Binary Operation '.' such that for any \(a, b \in G\), the equations \(a.x = b\) and \(y.a = b\) have unique solutions for \(x\) and \(y\) in G.

Let's check for the operation \(a.b = a - b\) on G:

For \(a.x = b\): \(a - x = b \implies x = a - b\). Since a and b are integers, \(a - b\) is a unique integer. So, a unique solution for x exists in G.

For \(y.a = b\): \(y - a = b \implies y = b + a\). Since a and b are integers, \(b + a\) is a unique integer. So, a unique solution for y exists in G.

Since the operation is a Binary Operation and satisfies the unique solvability property, the set of integer numbers G under subtraction is a quasi-group.

Conclusion Based on the Provided Options

We have established that the operation \(a.b = a - b\) on the set of integer numbers G:

• Is a Binary Operation (closure holds).
• Is not associative.
• Does not have an identity element.
• Therefore, is not a group.
• Satisfies the definition of a quasi-group.

Looking at the given options:

1. Binary Operation only
2. quasi-group
3. alone - group (This seems to refer to a group structure)
4. group

The operation is indeed a Binary Operation. It is also a quasi-group. It is not a group or 'alone - group'.

Among the given options, "Binary Operation only" highlights the most fundamental property that holds, distinguishing it from a full group structure. While it is also a quasi-group, the option "Binary Operation only" emphasizes that it satisfies the basic definition of a binary operation but fails the more stringent requirements for a group. Therefore, considering the options provided, "Binary Operation only" is the chosen description.