[size=150]An equivalent statement proof demonstrates that two statements are logically equivalent, meaning that they have the same truth value in all cases. In other words, if one statement is true, the other statement must also be true, and if one statement is false, the other statement must also be false. This type of proof involves establishing the bidirectional implication between the statements. [br][br]Example:[br]Claim: The statement "If x is an even number, then x[sup]2[/sup] is an even number" is equivalent to the statement "If x is an odd number, then x[sup]2[/sup] is an odd number."[br][br]Equivalent Statement Proof:[br]To prove the equivalence of the two statements, we need to show that each statement implies the other.[br][br]1. Statement 1: "If x is an even number, then x[sup]2[/sup] is an even number."[br][br]Assume x is an even number. We can represent this mathematically as x = 2k, where k is an integer. Substituting this into the equation x[sup]2[/sup], we have:[br]x[sup]2[/sup] = (2k)[sup]2[/sup] = 4k[sup]2[/sup].[br][br]Since k[sup]2[/sup] is an integer, 4k[sup]2[/sup] is an even number. Therefore, the statement "If x is an even number, then x[sup]2[/sup] is an even number" holds true.[br][br]2. Statement 2: "If x is an odd number, then x[sup]2[/sup] is an odd number."[br][br]Assume x is an odd number. We can represent this mathematically as x = 2k + 1, where k is an integer. Substituting this into the equation x^2, we have:[br]x[sup]2[/sup] = (2k + 1)[sup]2[/sup] = 4k[sup]2[/sup] + 4k + 1 = 2(2k[sup]2[/sup] + 2k) + 1.[br][br]Since 2k[sup]2[/sup] + 2k is an integer, 2(2k[sup]2[/sup] + 2k) + 1 is an odd number. Therefore, the statement "If x is an odd number, then x^2 is an odd number" also holds true.[br][br]Since we have shown that each statement implies the other, we can conclude that the two statements are equivalent.[br][br]In summary, an equivalent statement proof demonstrates that two statements have the same truth value in all cases. In this example, we proved that the statement "If x is an even number, then x[sup]2[/sup] is an even number" is equivalent to the statement "If x is an odd number, then x[sup]2[/sup] is an odd number" by showing the bidirectional implication between them using mathematical notation and logical reasoning.[/size]