Question

there are several statements in the table below. for each, determine whether it is a negation of this statement.
x is equal to 400.
|statement|negation?

it is not the case that x is equal to 400.	○	○
x is equal to 100.	○	○
it is not the case that x is equal to 500.	○	○
x is not equal to 400.	○	○

Explanation:

Step1: Recall Negation Definition

The negation of a statement \( P \) is a statement that is true when \( P \) is false and false when \( P \) is true. For the statement " \( x \) is equal to 400" (let's denote this as \( P \): \( x = 400 \)), its negation should be "it is not the case that \( x = 400 \)" or equivalently " \( x
eq 400 \)".

Step2: Analyze Each Statement

• Statement 1: "It is not the case that \( x \) is equal to 400."

This directly negates \( P \) (since it says \( P \) is not true), so it is a negation.

• Statement 2: " \( x \) is equal to 100."

This is a specific alternative value, not the logical negation of \( x = 400 \) (the negation of \( x = 400 \) allows \( x \) to be any value except 400, not just 100), so it is not a negation.

• Statement 3: "It is not the case that \( x \) is equal to 500."

This negates " \( x = 500 \)", not " \( x = 400 \)", so it is not a negation.

• Statement 4: " \( x \) is not equal to 400."

This is equivalent to "it is not the case that \( x = 400 \)", so it is a negation.

Answer:

(Filling the Table):

Statement	Negation? (Yes/No)
\( x \) is equal to 100.	No
It is not the case that \( x \) is equal to 500.	No
\( x \) is not equal to 400.	Yes

(To mark the circles: For the first and fourth statements, fill the "Yes" circle; for the second and third, fill the "No" circle.)