Question

Put the steps to the proof that the sum t=r+s
is irrational if r is a rational and s is rational in the correct sequence.

Step 1. Subtract, writing s as a fraction.

Step 2. For a contradiction, assume that t is rational, and write r and t as fractions.

Step 3. Realize a contradiction.

Step 4. Find a common denominator.

(1 point)
Responses

Step 4, Step 2, Step 1, Step 3
Step 4, Step 2, Step 1, Step 3

Step 3, Step 4, Step 2, Step 1
Step 3, Step 4, Step 2, Step 1

Step 1, Step 3, Step 2, Step 4
Step 1, Step 3, Step 2, Step 4

Step 2, Step 4, Step 1, Step 3

Answer 1

To prove that the sum \( t = r + s \) is irrational if \( r \) is rational and \( s \) is irrational, we can organize the steps in the correct sequence as follows:

1. **Step 2:** For a contradiction, assume that \( t \) is rational, and write \( r \) and \( t \) as fractions.
2. **Step 1:** Subtract, writing \( s \) as a fraction.
3. **Step 4:** Find a common denominator.
4. **Step 3:** Realize a contradiction.

Therefore, the correct order is **Step 2, Step 1, Step 4, Step 3**. This matches the option **Step 2, Step 4, Step 1, Step 3**.

To summarize, the reasoning involves first assuming \( t \) is rational, then manipulating the terms to ultimately reach a contradiction.