Consider the statement [tex]P[/tex]:
[tex]P. x=5[/tex]
Which of the following is an equivalent statement?
R: x+2=7
T: x \neq 5
Q: x is odd
S: x is a multiple of 5
Answer (2)
Solve for x in statement R: x + 2 = 7 A rr x = 5 .
Statement R is equivalent to statement P because it yields the same value for x.
Statements T, Q, and S are not equivalent to statement P because they either contradict it or do not uniquely define x.
The equivalent statement is R .
Explanation
Analyzing the Problem We are given the statement P : x = 5 and we need to find an equivalent statement from the given options. An equivalent statement means that the statement is true if and only if x = 5 . Let's analyze each option.
Analyzing Statement R Statement R : x + 2 = 7 . To check if this is equivalent to P , we solve for x in the equation x + 2 = 7 . Subtracting 2 from both sides, we get x = 7 − 2 = 5 . Thus, x = 5 , which is the same as statement P . So, statement R is equivalent to statement P .
Analyzing Statement T Statement T : x e q 5 . This statement says that x is not equal to 5. This is the opposite of statement P , so it is not equivalent.
Analyzing Statement Q Statement Q : x is odd. We know that x = 5 , and 5 is an odd number. However, this statement is not equivalent to P because there are other odd numbers besides 5 (e.g., 1, 3, 7, 9, etc.). So, if x is odd, it does not necessarily mean that x = 5 .
Analyzing Statement S Statement S : x is a multiple of 5. We know that x = 5 , and 5 is a multiple of 5. However, this statement is not equivalent to P because there are other multiples of 5 besides 5 (e.g., 10, 15, 20, etc.). So, if x is a multiple of 5, it does not necessarily mean that x = 5 .
Conclusion Therefore, the only statement equivalent to P : x = 5 is R : x + 2 = 7 .
Examples
In programming, you might use equivalent conditions to check if a certain value meets a specific criterion. For example, if you want to check if a user's age is 18, you could use the condition 'age == 18' or an equivalent condition like 'age + 2 == 20'. Both conditions achieve the same result, ensuring that the program behaves as expected regardless of which condition is used. This is also useful in simplifying logical expressions in circuit design or in database queries where multiple conditions can be expressed in different but equivalent ways to optimize performance.
The equivalent statement to P : x = 5 is R : x + 2 = 7 because solving R leads to x = 5 . Statements T , Q , and S do not uniquely define x as equal to 5. Thus, the correct choice is R .
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