Determine the truth value of the inverse. Explain.
Answer (1)
The original statement is: If x ≤ − 500 , then x < 500 .
The inverse statement is: If x ≥ 500 , then -500"> x > − 500 .
Evaluate the truth value of the inverse statement: If x ≥ 500 , then -500"> x > − 500 is always true.
Conclude that the inverse statement is true: T r u e .
Explanation
Analyze the statement Let's analyze the given statement and its inverse to determine the truth value.
State the original statement The original statement is: If a number is not greater than -500, then it is less than 500. We can write this as: If x ≤ − 500 , then x < 500 .
State the inverse statement The inverse of the statement is: If a number is not less than 500, then it is not greater than -500. We can write this as: If x ≥ 500 , then -500"> x > − 500 .
Evaluate the truth value Now, let's evaluate the truth value of the inverse statement. If x is greater than or equal to 500, is it always greater than -500? Yes, any number greater than or equal to 500 is also greater than -500.
Conclusion Therefore, the inverse statement is true.
Examples
Understanding inverse statements is crucial in logic and mathematics. For instance, consider a security system: 'If the alarm is triggered, then there was an intrusion.' The inverse would be: 'If the alarm is not triggered, then there was no intrusion.' Evaluating the truth of such inverses helps in assessing the reliability of the system. In this case, if the alarm isn't triggered, it's highly probable there was no intrusion, but there could be other possibilities, such as a system malfunction or a silent entry, which highlights the importance of carefully analyzing inverse statements.
