Which of the following is a biconditional statement?
A) If [tex]$x \neq 5$[/tex] then [tex]$x^2 \neq 25$[/tex]
B) [tex]$x=5$[/tex] if and only if [tex]$x+5=10$[/tex]
C) [tex]$x=5$[/tex] if [tex]$x^2=25$[/tex]
D) If [tex]$x^2=25$[/tex], then [tex]$x=5$[/tex] or [tex]$x=-5$[/tex]
Answer (1)
A biconditional statement is true if both parts are true or both are false.
Option A is false because x = − 5 satisfies x e q 5 but x 2 e q 25 is false.
Option B, x = 5 if and only if x + 5 = 10 , is true in both directions.
Option C is false because if x 2 = 25 , x could be − 5 .
Therefore, the biconditional statement is B .
Explanation
Understanding Biconditional Statements We need to identify the biconditional statement among the given options. A biconditional statement is true if and only if both parts are true or both parts are false.
Analyzing Option A Option A: If x e q 5 then x 2 e q 25 . This is false because if x = − 5 , then x e q 5 is true, but x 2 e q 25 is false since ( − 5 ) 2 = 25 .
Analyzing Option B Option B: x = 5 if and only if x + 5 = 10 . If x = 5 , then x + 5 = 5 + 5 = 10 , which is true. If x + 5 = 10 , then subtracting 5 from both sides gives x = 5 , which is also true. Thus, this is a biconditional statement.
Analyzing Option C Option C: x = 5 if x 2 = 25 . This can be rewritten as: If x 2 = 25 then x = 5 . This is false because x could be − 5 , since ( − 5 ) 2 = 25 .
Analyzing Option D Option D: If x 2 = 25 , then x = 5 or x = − 5 . This is true, but it is not a biconditional statement. It is a conditional statement.
Conclusion Therefore, the biconditional statement is option B.
Examples
Biconditional statements are used in various fields, such as logic, mathematics, and computer science. For example, in defining necessary and sufficient conditions. Consider a scenario where a light switch must be on for a light bulb to illuminate. The statement 'The light is on if and only if the switch is on' is a biconditional statement. This means that if the light is on, the switch must be on, and if the switch is on, the light must be on. This concept is crucial in understanding cause-and-effect relationships and logical equivalences.
