Negation of the given statement p is
P : There exists a rational number y such that `y^3 ne 108`

Options

A. There exists an irrational number y such that `y^3 = 108`
B. There exists an irrational number y such that `y^3 ne 108`
C. There does not exist a rational number such `y^3 = 108`
D. There does not exist a rational number such `y^3 ne 108`

Answer

To find the negation of the given statement \( p \), we will follow these steps:

### Step 1: Understand the original statement
The original statement \( p \) is:
"There exists a rational number \( y \) such that \( y^3 \neq 108 \)".

### Step 2: Identify the logical structure
The statement uses the existential quantifier "there exists", which can be represented in logical notation as:
\[ p: \exists y \in \mathbb{Q}, \, y^3 \neq 108 \]

### Step 3: Apply the negation
The negation of an existential statement \( \exists \) is a universal statement \( \forall \). Therefore, the negation of \( p \) will be:
\[ \neg p: \forall y \in \mathbb{Q}, \, y^3 = 108 \]

### Step 4: Write the final negated statement
The negated statement can be expressed in words as:
"For all rational numbers \( y \), \( y^3 \) is equal to 108".

Thus, the negation of the given statement \( p \) is:
"There does not exist a rational number \( y \) such that \( y^3 \neq 108 \)" or equivalently, "For all rational numbers \( y \), \( y^3 = 108 \)".

### Summary of the solution:
- Original statement: "There exists a rational number \( y \) such that \( y^3 \neq 108 \)".
- Negated statement: "For all rational numbers \( y \), \( y^3 = 108 \)".

---