Question

Choose two statements (the original conditional statement and its converse) that could form the biconditional ""x = 1 if and only if x - 1 = 0."" 0 A. If x-1 = 0, then x = 1. B. If x= 1, then x-10. C. If x = 1, then x-1 = 0. m D. If x = 1, then x-1 +0.

Answer 1

The two statements that form the biconditional "x = 1 if and only if x - 1 = 0" are: Statement A (If x-1 = 0, then x = 1) and Statement C (If x = 1, then x-1 = 0), expressing necessary and sufficient conditions.

Step-by-step explanation:

The question asks to identify the two statements that could form the biconditional "x = 1 if and only if x - 1 = 0." A biconditional statement is true when both the original conditional statement and its converse are true. This means we are looking for one statement where x = 1 is the antecedent (or sufficient condition) and x - 1 = 0 is the consequent (or necessary condition), and another statement where these conditions are flipped.

The correct statements that form this biconditional are:

1. A. If x-1 = 0, then x = 1.
2. C. If x = 1, then x-1 = 0.

Statement A expresses that it is sufficient to have x-1 = 0 for x to be 1, which makes x-1 = 0 the antecedent and x = 1 the consequent. Statement C expresses that if x = 1 (the antecedent), then necessarily x-1 = 0 (the consequent). Here, x = 1 is the sufficient condition for x-1 = 0 to occur. Together, these two statements express the necessary and sufficient conditions that define the biconditional relationship.