Which is a counterexample for the assertion $(a+b)^2=a^2+b^2$?
A. $a=0, b=1$
B. There is no counterexample because this assertion is true.
C. $a=1, b=1$
D. $a=1, b=0
Answer (2)
Test a = 0 , b = 1 : ( 0 + 1 ) 2 = 1 and 0 2 + 1 2 = 1 . Not a counterexample.
Test a = 1 , b = 1 : ( 1 + 1 ) 2 = 4 and 1 2 + 1 2 = 2 . This is a counterexample.
Test a = 1 , b = 0 : ( 1 + 0 ) 2 = 1 and 1 2 + 0 2 = 1 . Not a counterexample.
The counterexample is a = 1 , b = 1 , so the answer is a = 1 , b = 1 .
Explanation
Understanding the Problem We are given the assertion ( a + b ) 2 = a 2 + b 2 and asked to find a counterexample from the given options. A counterexample is a pair of values for a and b for which the equation is false.
Testing a=0, b=1 Let's test the first option, a = 0 and b = 1 . We have ( 0 + 1 ) 2 = 1 2 = 1 and 0 2 + 1 2 = 0 + 1 = 1 . Since ( 0 + 1 ) 2 = 0 2 + 1 2 , this is not a counterexample.
Testing a=1, b=1 Next, let's test the option a = 1 and b = 1 . We have ( 1 + 1 ) 2 = 2 2 = 4 and 1 2 + 1 2 = 1 + 1 = 2 . Since ( 1 + 1 ) 2 e q 1 2 + 1 2 , this is a counterexample.
Testing a=1, b=0 Finally, let's test the option a = 1 and b = 0 . We have ( 1 + 0 ) 2 = 1 2 = 1 and 1 2 + 0 2 = 1 + 0 = 1 . Since ( 1 + 0 ) 2 = 1 2 + 0 2 , this is not a counterexample.
Conclusion Therefore, the counterexample is a = 1 and b = 1 .
Examples
Understanding counterexamples is crucial in various fields. For instance, in engineering, when designing a bridge, engineers use mathematical models to predict the bridge's behavior under different loads. If a counterexample is found, it means the model is flawed and needs refinement to ensure the bridge's safety. Similarly, in computer science, counterexamples are used to test the correctness of algorithms and software. By identifying scenarios where the algorithm fails, developers can improve its reliability and robustness. This process ensures that the software functions correctly under a wide range of conditions, preventing potential errors and vulnerabilities.
The counterexample for the assertion ( a + b ) 2 = a 2 + b 2 is found using the values a = 1 and b = 1 , resulting in ( 1 + 1 ) 2 = 1 2 + 1 2 . The correct choice is therefore a = 1 , b = 1 .
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