Use a quantifier to make the following true or false, as indicated, where x is a natural number.
-$x - 13 = 13 - x$ (make false)
A) There exists a natural number $x$ such that $x - 13 = 13 - x$.
B) At least one natural number $x$ exists such that $x - 13 = 13 - x$.
C) There is no natural number $x$ such that $x - 13 = 13 - x$.
D) For $x = 13 , x - 13 = 13 - x$.

Question

Quizplus · Accepted Answer

The equation $x - 13 = 13 - x$ simplifies to $2x = 26$, which has the solution $x = 13$. However, the statement to make it false with a quantifier is "There is no natural number $x$ such that $x - 13 = 13 - x$," because the question asks for a quantifier that makes the statement false, and option C correctly asserts that no such natural number exists under the false-making condition. However, the explanation contradicts the selection because $x = 13$ is indeed a solution, making the equation true. The correct interpretation of making the statement false would involve asserting a condition that contradicts the existence of a solution, but in this context, the explanation provided does not align with the selection. The correct approach to making the statement false would be to assert a condition that no natural number satisfies the equation under the given or a different context, but given the equation's validity for $x = 13$, option C's assertion is inherently true, not false. The confusion arises from the misinterpretation of the task, which involves selecting a statement that would be inherently false given the equation's properties, but the explanation mistakenly validates the equation while selecting an option that implies its invalidity for all natural numbers.

Use a Quantifier to Make the Following True or False x−13=13−xx - 13 = 13 - xx−13=13−x

Use a Quantifier to Make the Following True or False $x - 13 = 13 - x$