Placement Of "for All N" In "show That A(n) Holds"

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Introduction

In mathematical writing, it is crucial to use precise language to convey complex ideas. One common phrase used in mathematical statements is "for all n." However, when combining this phrase with the statement "show that A(n) holds," the placement of "for all n" can be unclear. In this article, we will explore the correct placement of "for all n" in the sentence "show that A(n) holds" and provide guidelines for grammatical correctness.

Understanding the Statement

Before we dive into the placement of "for all n," let's break down the statement "show that A(n) holds." This statement is asking the reader to prove that the mathematical expression A(n) is true for a given value of n. The phrase "for all n" indicates that the statement A(n) is true for all values of n, not just a specific value.

Placement of "for all n"

The placement of "for all n" in the sentence "show that A(n) holds" can be either before or after the statement A(n). However, the correct placement depends on the context and the intended meaning.

Before A(n)

When "for all n" is placed before A(n), it means that the statement A(n) is true for all values of n. This placement is often used in mathematical proofs to establish a general result.

Example: "Show that for all n, A(n) holds."

In this example, the phrase "for all n" is placed before A(n), indicating that the statement A(n) is true for all values of n.

After A(n)

When "for all n" is placed after A(n), it means that the statement A(n) is true for a specific value of n. This placement is often used in mathematical statements to specify a particular value of n.

Example: "Show that A(n) holds for all n."

In this example, the phrase "for all n" is placed after A(n), indicating that the statement A(n) is true for all values of n.

Guidelines for Placement

To ensure grammatical correctness, follow these guidelines for the placement of "for all n" in the sentence "show that A(n) holds":

  • When establishing a general result, place "for all n" before A(n).
  • When specifying a particular value of n, place "for all n" after A(n).
  • Avoid placing "for all n" in the middle of the sentence, as this can lead to ambiguity.

Conclusion

In conclusion, the placement of "for all n" in the sentence "show that A(n) holds" is crucial for grammatical correctness. By following the guidelines outlined in this article, you can ensure that your mathematical statements are clear and concise. Remember to place "for all n" before A(n) to establish a general result, and after A(n) to specify a particular value of n.

Additional Tips

  • When writing mathematical statements, use precise language to avoid ambiguity.
  • Use parentheses to clarify the scope of the statement.
  • Avoid using ambiguous phrases, such as "for all n" in the middle of the sentence.

Common Mistakes

  • Placing "for all n" in the middle of the sentence, leading to ambiguity.
  • Failing to use precise language, resulting in statements.
  • Not using parentheses to clarify the scope of the statement.

Best Practices

  • Use "for all n" before A(n) to establish a general result.
  • Use "for all n" after A(n) to specify a particular value of n.
  • Avoid using ambiguous phrases and use precise language instead.

Conclusion

In conclusion, the placement of "for all n" in the sentence "show that A(n) holds" is a crucial aspect of mathematical writing. By following the guidelines outlined in this article, you can ensure that your mathematical statements are clear, concise, and grammatically correct. Remember to use precise language, avoid ambiguity, and use parentheses to clarify the scope of the statement.

Introduction

In our previous article, we explored the correct placement of "for all n" in the sentence "show that A(n) holds." However, we understand that there may be additional questions and concerns regarding this topic. In this article, we will address some of the most frequently asked questions about the placement of "for all n" in mathematical statements.

Q&A

Q: What is the difference between "for all n" and "for some n"?

A: For all n means that the statement A(n) is true for all values of n, while for some n means that the statement A(n) is true for at least one value of n.

Example: "Show that for all n, A(n) holds" vs. "Show that for some n, A(n) holds."

Q: Can I place "for all n" in the middle of the sentence?

A: No, it is generally not recommended to place "for all n" in the middle of the sentence, as this can lead to ambiguity. Instead, place it before or after A(n) to ensure clarity.

Example: "Show that A(n) holds for all n" is better than "Show that A(n) holds for all n, but not for some n."

Q: How do I know when to use "for all n" and when to use "for some n"?

A: Use "for all n" when you want to establish a general result that applies to all values of n. Use "for some n" when you want to specify a particular value of n or when the statement A(n) is true for at least one value of n.

Example: "Show that for all n, A(n) holds" (general result) vs. "Show that for some n, A(n) holds" (specific value of n).

Q: Can I use "for all n" with other quantifiers, such as "there exists"?

A: Yes, you can use "for all n" with other quantifiers, such as "there exists." However, be careful to use parentheses to clarify the scope of the statement.

Example: "Show that for all n, there exists a value of m such that A(n) holds."

Q: How do I handle multiple quantifiers in a single statement?

A: Use parentheses to clarify the scope of each quantifier. This will help to avoid ambiguity and ensure that the statement is clear.

Example: "Show that for all n, there exists a value of m such that A(n) holds."

Q: Can I use "for all n" with a specific value of n?

A: Yes, you can use "for all n" with a specific value of n. However, be careful to use parentheses to clarify the scope of the statement.

Example: "Show that for all n, A(n) holds, where n = 5."

Q: How do I handle negations with "for all n"?

A: Use the negation symbol to indicate that the statement A(n) is not true for all values of n.

Example: "Show that not for all n, A(n) holds."

Conclusion

In conclusion, the placement of "for all n" in the sentence "show that A(n) holds" is a crucial aspect of mathematical writing. By following the guidelines outlined in this article and addressing common questions and concerns, you can ensure that your mathematical statements are clear, concise, and grammatically correct.

Additional Tips

  • Use precise language to avoid ambiguity.
  • Use parentheses to clarify the scope of the statement.
  • Avoid using ambiguous phrases and use precise language instead.

Common Mistakes

  • Placing "for all n" in the middle of the sentence, leading to ambiguity.
  • Failing to use precise language, resulting in statements.
  • Not using parentheses to clarify the scope of the statement.

Best Practices

  • Use "for all n" before A(n) to establish a general result.
  • Use "for all n" after A(n) to specify a particular value of n.
  • Avoid using ambiguous phrases and use precise language instead.
  • Use parentheses to clarify the scope of the statement.