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 has the form "P if and only if Q".
Option A is a conditional statement and is false.
Option B is a biconditional statement and is true: x = 5 if and only if x + 5 = 10 .
Option C is a conditional statement and is false.
Option D is a conditional statement.
The correct answer is B .
Explanation
Understanding Biconditional Statements A biconditional statement is a statement that is true if and only if both parts are true or both parts are false. It has the form 'P if and only if Q', meaning that P implies Q and Q implies P. We need to identify which of the given options fits this definition.
Analyzing Option A Option A: 'If x e q 5 then x 2 e q 25 '. This is a conditional statement. If x = − 5 , then x e q 5 is true, but x 2 e q 25 is false, since ( − 5 ) 2 = 25 . Thus, this statement is false.
Analyzing Option B Option B: ' x = 5 if and only if x + 5 = 10 '. This is a biconditional statement. If x = 5 , then x + 5 = 10 . If x + 5 = 10 , then x = 5 . This statement is true.
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 a conditional statement. If x = − 5 , then x 2 = 25 is true, but x = 5 is false. Thus, this statement is false.
Analyzing Option D Option D: 'If x 2 = 25 , then x = 5 or x = − 5 '. This is a conditional statement. If x 2 = 25 , then x = 5 or x = − 5 is true. However, the reverse, 'If x = 5 or x = − 5 , then x 2 = 25 ' is also true. While the statement itself is true, it is not in the simple biconditional form 'P if and only if Q' where P and Q are simple statements.
Conclusion Therefore, the correct answer is option B, which is a biconditional statement in the form 'P if and only if Q'.
Examples
Biconditional statements are used in various fields, such as mathematics and computer science, to define equivalences. For example, in geometry, we can say that a triangle is equilateral if and only if all its angles are equal. This means that if a triangle is equilateral, then all its angles are equal, and if all the angles of a triangle are equal, then the triangle is equilateral. Biconditional statements help establish clear and precise definitions and relationships.
