write in symbolic Language, negate, it, and determine if statement is true or False with a proof D) There is a natural number s such that for any natural number n s is the product of n with itself. negation (FSEN) (V₁EN) (S = 1²) (USEN) (FREN) (s+₁²)
![**Title: Understanding Symbolic Logic and Negation in Mathematics**
**Objective:**
Learn how to write statements in symbolic language, negate them, and determine their truth value with a proof.
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**Statement to Analyze:**
**D) Original Statement:**
"There is a natural number \( s \) such that \( s \) is the product of \( n \) with itself."
**Symbolic Representation:**
\[
\exists s \in \mathbb{N} (s = n^2)
\]
**Negation:**
The negation of the statement involves stating the opposite, which is formulated as:
\[
\forall n \in \mathbb{N} \neg(s = n^2)
\]
**Interpretation of the Negation:**
"For any natural number \( n \), there is no \( s \) such that \( s = n^2 \)."
**Truth Value:**
Investigate whether the negation holds true or false, supported by a proof. In this context, the assertion of the negation would be false, as any natural number squared results in an \( s \) meeting the original condition.
**Conclusion:**
The exercise demonstrates how to handle logical statements in mathematics, particularly focusing on negation and the evaluation of their truthfulness through symbolic representation.
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This breakdown aids in grasping essential concepts in logic that are pivotal in mathematical reasoning.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2F39d575d2-518e-498e-807c-f6884eff5ab4%2F2rybw6d_processed.jpeg&w=3840&q=75)
What is Statement:
A sentence that isn't an order, an exclamation mark, or a question is often referred to as a statement. Generally, a statement can be classified as either always true, always false, or ambiguous. In mathematics, a proposition is only acceptable or valid if it is either true or false. This kind of statement is referred to as a mathematically admissible assertion. To put it another way, a statement is true if it holds regardless of the situation; otherwise, it is false. Simple and complex statements are the two different categories of statements.
Given:
The given statement is there is a natural number such that for any natural number , is the product of with itself.
To Determine:
We symbolically represent the given statement. Then, we determine its negation and its truth value.
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