Write down the negation of following compound statement: |x| is equal to x or -x.

Answer

To find the negation of the compound statement "|x| is equal to x or -x", we will follow these steps:

### Step 1: Understand the Original Statement
The original statement is:
\[ |x| = x \text{ or } |x| = -x \]

### Step 2: Rewrite the Statement in Logical Form
We can express the original statement in logical terms:
Let \( P \) be the statement \( |x| = x \) and \( Q \) be the statement \( |x| = -x \).
The original statement can be written as:
\[ P \lor Q \]
where \( \lor \) represents the logical "or".

### Step 3: Apply Negation to the Compound Statement
The negation of a compound statement in the form \( P \lor Q \) is given by:
\[ \neg (P \lor Q) = \neg P \land \neg Q \]
This means we need to negate both parts of the statement and change "or" to "and".

### Step 4: Negate Each Component
Now we negate each part:
- The negation of \( P \) (i.e., \( |x| = x \)) is:
\[ \neg P: |x| \neq x \]
- The negation of \( Q \) (i.e., \( |x| = -x \)) is:
\[ \neg Q: |x| \neq -x \]

### Step 5: Combine the Negations
Now, we combine the negations using "and":
\[ \neg (P \lor Q) = \neg P \land \neg Q \]
Thus, we have:
\[ |x| \neq x \land |x| \neq -x \]

### Final Statement
The negation of the original compound statement is:
\[ |x| \neq x \text{ and } |x| \neq -x \]

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