Question

Suppose someone plans to write an indirect proof of the conditional. Write a correct first sentence of the indirect proof.

If $x^2\ne y^2$, then $x\ne y$.

Short answer

The correct first sentence would be “Assume that $x=y$.”

Step-by-step solution

Step 1. Apply the concept of indirect proof.

The indirect proof is based on the classical notion that any given sentence, such as the conclusion, must be either true or false. The Indirect proof is done by assuming the premises to be true and the conclusion to be false and deriving a contradiction.

Step 2. Step description.

Consider the statement.

If $x^2\ne y^2$then $x\ne y$.

If someone wants to write an indirect proof for this statement then there is a need to write a correct first sentence of this indirect proof assuming the conclusion of the proof is not true.

Step 3. Step description.

The correct first sentence of the indirect proof is, Assume that $x=y$.

Therefore, the correct first sentence of this indirect proof is “Assume that $x=y$.”