E) Prove That If A + C = B + C A + C = B + C A + C = B + C , Then A = B A = B A = B When A , B , C ∈ R A, B, C \in \mathbb{R} A , B , C ∈ R .

Mar 9, 2025 by ADMIN 142 views

**Proving the Equality of Real Numbers**

Introduction

In mathematics, proving the equality of real numbers is a fundamental concept that forms the basis of various mathematical operations and theorems. One such theorem states that if $a + c = b + c$ , then $a = b$ when $a, b, c \in \mathbb{R}$ . In this article, we will delve into the proof of this theorem and explore its significance in the realm of real numbers.

The Theorem

The theorem in question can be stated as follows:

If $a + c = b + c$ , then $a = b$ when $a, b, c \in \mathbb{R}$ .

Proof

To prove this theorem, we will employ a simple yet effective method. We will start by assuming that the given equation holds true, and then we will manipulate the equation to arrive at the desired conclusion.

Step 1: Assume the Given Equation

Let's assume that $a + c = b + c$ . This is the given equation that we will work with.

Step 2: Subtract c from Both Sides

Next, we will subtract $c$ from both sides of the equation to isolate the terms involving $a$ and $b$ .

$a + c - c = b + c - c$

Simplifying the equation, we get:

$a = b$

Step 3: Conclusion

We have now arrived at the desired conclusion, which states that $a = b$ when $a, b, c \in \mathbb{R}$ .

Significance of the Theorem

The theorem we have just proved has significant implications in the realm of real numbers. It shows that if two real numbers are added to the same third real number, then the two original numbers are equal. This theorem forms the basis of various mathematical operations and theorems, including the concept of equality and the properties of real numbers.

Real-World Applications

The theorem we have proved has numerous real-world applications in various fields, including:

Finance: In finance, the theorem can be used to compare the values of two different investments. If two investments have the same return when added to a third investment, then the two original investments are equal.
Science: In science, the theorem can be used to compare the values of two different physical quantities. If two physical quantities have the same value when added to a third physical quantity, then the two original quantities are equal.
Engineering: In engineering, the theorem can be used to compare the values of two different engineering quantities. If two engineering quantities have the same value when added to a third engineering quantity, then the two original quantities are equal.

Conclusion

In conclusion, the theorem we have proved states that if $a + c = b + c$ , then $a = b$ when $a, b, c \in \mathbb{R}$ . This theorem has significant implications in the realm of real numbers and has numerous real-world applications in various fields. We hope that this article has provided a clear and concise proof of the theorem and has helped to illustrate its significance and importance.

References

[1] "Real Numbers" by Math Open Reference
[2] "Equality of Real Numbers" by Khan Academy
[3] "Properties of Real Numbers" by Wolfram MathWorld

Glossary

Real Numbers: A set of numbers that includes all rational and irrational numbers.
Equality: A mathematical operation that states that two or more quantities are equal.
Properties of Real Numbers: A set of rules that govern the behavior of real numbers.

FAQs

Q: What is the theorem in question? A: The theorem states that if $a + c = b + c$ , then $a = b$ when $a, b, c \in \mathbb{R}$ .
Q: What is the significance of the theorem? A: The theorem has significant implications in the realm of real numbers and has numerous real-world applications in various fields.
Q: How is the theorem used in real-world applications? A: The theorem is used to compare the values of two different quantities, whether they be financial, scientific, or engineering quantities.
Frequently Asked Questions (FAQs) =====================================

Q: What is the theorem in question?

A: The theorem states that if $a + c = b + c$ , then $a = b$ when $a, b, c \in \mathbb{R}$ .

Q: What is the significance of the theorem?

A: The theorem has significant implications in the realm of real numbers and has numerous real-world applications in various fields.

Q: How is the theorem used in real-world applications?

A: The theorem is used to compare the values of two different quantities, whether they be financial, scientific, or engineering quantities.

Q: What are some examples of real-world applications of the theorem?

A: Some examples of real-world applications of the theorem include:

Comparing the values of two different investments in finance
Comparing the values of two different physical quantities in science
Comparing the values of two different engineering quantities in engineering

Q: How is the theorem proved?

A: The theorem is proved by assuming that the given equation holds true, and then manipulating the equation to arrive at the desired conclusion.

Q: What are the steps involved in proving the theorem?

A: The steps involved in proving the theorem are:

Assume the given equation
Subtract c from both sides
Simplify the equation
Arrive at the desired conclusion

Q: What are the implications of the theorem?

A: The implications of the theorem are that if two real numbers are added to the same third real number, then the two original numbers are equal.

Q: Can the theorem be applied to other mathematical operations?

A: Yes, the theorem can be applied to other mathematical operations, such as multiplication and division.

Q: How does the theorem relate to other mathematical concepts?

A: The theorem relates to other mathematical concepts, such as equality and the properties of real numbers.

Q: What are some common misconceptions about the theorem?

A: Some common misconceptions about the theorem include:

Thinking that the theorem only applies to addition
Thinking that the theorem only applies to real numbers
Thinking that the theorem is only used in simple mathematical operations

Q: How can the theorem be used to solve real-world problems?

A: The theorem can be used to solve real-world problems by comparing the values of two different quantities, whether they be financial, scientific, or engineering quantities.

Q: What are some tips for applying the theorem in real-world situations?

A: Some tips for applying the theorem in real-world situations include:

Always assuming that the given equation holds true
Always manipulating the equation to arrive at the desired conclusion
Always considering the implications of the theorem

Q: What are some resources for further learning about the theorem?

A: Some resources for further learning about the theorem include:

"Real Numbers" by Math Open Reference
"Equality of Real Numbers" by Khan Academy
"Properties of Real Numbers" by Wolfram MathWorld

Q: How can the theorem be used to improve mathematical understanding?

A: The theorem can be used to improve mathematical understanding by providing a clear and concise proof of the theorem and by illustrating its significance and importance.

Q: What are some common mistakes to avoid when applying the theorem?

A: Some common mistakes to avoid when applying the theorem include:

Not assuming that the given equation holds true
Not manipulating the equation to arrive at the desired conclusion
Not considering the implications of the theorem

Q: How can the theorem be used to improve problem-solving skills?

A: The theorem can be used to improve problem-solving skills by providing a clear and concise method for solving problems and by illustrating its significance and importance.