The Operation { \ \textless \ Em\ \textgreater \ $}$ Is Defined By { A \ \textless \ /em\ \textgreater \ B = \frac{1}{2}(a - B)$}$ For { A \neq B$} . A ) I ) F O R M A T A B L E O F T H E O P E R A T I O N F O R T H E S E T \[ .a) I) Form A Table Of The Operation For The Set \[ . A ) I ) F Or Ma T Ab L Eo F T H Eo P Er A T I O N F Or T H Ese T \[ {4, 6,

Introduction

In mathematics, various operations are defined to perform specific tasks. One such operation is defined as , which is used to combine two numbers. In this article, we will explore the properties of this operation and form a table for the set {4, 6, 8, 10, 12}.

Definition of the Operation

The operation is defined as:

$$a \ \textless \ /em\ \textgreater \ b = \frac{1}{2}(a - b)$$

for $a \neq b$.

Properties of the Operation

To understand the properties of the operation, let's first calculate the result of for different pairs of numbers.

Commutative Property

The commutative property states that the order of the numbers does not change the result.

$a$	$b$	$a \ \textless \ /em\ \textgreater \ b$	$b \ \textless \ /em\ \textgreater \ a$
4	6	2.0	2.0
6	4	2.0	2.0
8	10	4.0	4.0
10	8	4.0	4.0
12	6	6.0	6.0

As we can see, the result of is the same for both $a \ \textless \ /em\ \textgreater \ b$ and $b \ \textless \ /em\ \textgreater \ a$, which means the operation is commutative.

Associative Property

The associative property states that the order in which we perform the operation does not change the result.

$a$	$b$	$c$	$(a \ \textless \ /em\ \textgreater \ b) \ \textless \ /em\ \textgreater \ c$	$a \ \textless \ /em\ \textgreater \ (b \ \textless \ /em\ \textgreater \ c)$
4	6	8	5.0	5.0
6	4	8	5.0	5.0
8	10	12	8.0	8.0
10	8	12	8.0	8.0
12	6	8	10.0	10.0

As we can see, the result of is the same for both $(a \ \textless \ /em\ \textgreater \ b) \ \textless \ /em\ \textgreater \ c$ and $a \ \textless \ /em\ \textgreater \ (b \ \textless \ /em\ \textgreater \ c)$, which means the operation is associative.

Distributive Property

The distributive property states that the operation can be distributed over addition.

$a$	$b$	$c$	$(a + b) \ \textless \ /em\ \textgreater \ c$	$a \ \textless \ /em\ \textgreater \ b + a \ \textless \ /em\ \textgreater \ c$
4	6	8	6.0	5.0
6	4	8	6.0	5.0
8	10	12	10.0	8.0
10	8	12	10.0	8.0
12	6	8	12.0	10.0

As we can see, the result of is not the same for both $(a + b) \ \textless \ /em\ \textgreater \ c$ and $a \ \textless \ /em\ \textgreater \ b + a \ \textless \ /em\ \textgreater \ c$, which means the operation is not distributive.

Identity Element

The identity element is an element that does not change the result when combined with another element.

$a$	$b$	$a \ \textless \ /em\ \textgreater \ b$	$a \ \textless \ /em\ \textgreater \ e$	$e \ \textless \ /em\ \textgreater \ a$
4	6	2.0	4.0	4.0
6	4	2.0	4.0	4.0
8	10	4.0	8.0	8.0
10	8	4.0	8.0	8.0
12	6	6.0	12.0	12.0

As we can see, the result of is not the same for both $a \ \textless \ /em\ \textgreater \ e$ and $e \ \textless \ /em\ \textgreater \ a$, which means the operation does not have an identity element.

Inverse Element

The inverse element is an element that, when combined with another element, results in the identity element.

$a$	$b$	$a \ \textless \ /em\ \textgreater \ b$	$a \ \textless \ /em\ \textgreater \ a^{-1}$	$a^{-1} \ \textless \ /em\ \textgreater \ a$
4	6	2.0	4.0	4.0
6	4	2.0	4.0	4.0
8	10	4.0	8.0	8.0
10	8	4.0	8.0	8.0
12	6	6.0	12.0	12.0

As we can see, the result of is not the same for both $a \ \textless \ /em\ \textgreater \ a^{-1}$ and $a^{-1} \ \textless \ /em\ \textgreater \ a$, which means the operation does not have an inverse element.

Conclusion

In conclusion, the operation has the following properties:

• Commutative: Yes
• Associative: Yes
• Distributive: No
• Identity Element: No
• Inverse Element: No

Q: What is the operation and how is it defined?

A: The operation is defined as:

$$a \ \textless \ /em\ \textgreater \ b = \frac{1}{2}(a - b)$$

for $a \neq b$.

Q: What are the properties of the operation ?

A: The operation has the following properties:

• Commutative: Yes
• Associative: Yes
• Distributive: No
• Identity Element: No
• Inverse Element: No

Q: Is the operation commutative?

A: Yes, the operation is commutative, which means that the order of the numbers does not change the result.

Q: Is the operation associative?

A: Yes, the operation is associative, which means that the order in which we perform the operation does not change the result.

Q: Is the operation distributive?

A: No, the operation is not distributive, which means that it cannot be distributed over addition.

Q: Does the operation have an identity element?

A: No, the operation does not have an identity element, which means that there is no element that does not change the result when combined with another element.

Q: Does the operation have an inverse element?

A: No, the operation does not have an inverse element, which means that there is no element that, when combined with another element, results in the identity element.

Q: Can the operation be used to solve equations?

A: Yes, the operation can be used to solve equations, but it may not always be the most efficient or effective method.

Q: Can the operation be used in real-world applications?

A: Yes, the operation can be used in real-world applications, such as in finance, engineering, and science.

Q: Is the operation a binary operation?

A: Yes, the operation is a binary operation, which means that it takes two inputs and produces a single output.

Q: Can the operation be used with negative numbers?

A: Yes, the operation can be used with negative numbers, but the result may be a negative number.

Q: Can the operation be used with fractions?

A: Yes, the operation can be used with fractions, but the result may be a fraction.

Q: Can the operation be used with decimals?

A: Yes, the operation can be used with decimals, but the result may be a decimal.

Conclusion

In conclusion, the operation is a binary operation that combines two numbers to produce a result. It has the commutative and associative properties, but it does not have the distributive, identity element, or inverse element properties. The operation can be used to solve equations and in real-world applications, but it may not always be the most efficient or effective method.