Write the contrapositive of the following statements: If x is real number such that `0 lt x lt 1`, then `x^(2) lt 1`.

Answer

To find the contrapositive of the given statement, we can follow these steps:

### Step 1: Understand the Original Statement
The original statement is:
"If \( x \) is a real number such that \( 0 < x < 1 \), then \( x^2 < 1 \)."

### Step 2: Identify the Hypothesis and Conclusion
- **Hypothesis (P)**: \( 0 < x < 1 \)
- **Conclusion (Q)**: \( x^2 < 1 \)

### Step 3: Write the Contrapositive
The contrapositive of a statement of the form "If P, then Q" is "If not Q, then not P."

- **Not Q**: \( x^2 \geq 1 \) (This means \( x^2 \) is either greater than or equal to 1)
- **Not P**: \( x \) is not a real number such that \( 0 < x < 1 \) (This means \( x \) is either less than or equal to 0 or greater than or equal to 1)

### Step 4: Combine the Contrapositive
Now, we can combine these to form the contrapositive statement:
"If \( x^2 \geq 1 \), then \( x \) is not a real number such that \( 0 < x < 1 \)."

### Final Answer
The contrapositive of the given statement is:
"If \( x^2 \geq 1 \), then \( x \) is not a real number such that \( 0 < x < 1 \)."

---