Question

Show that the statement “For any real numbers a and b , a 2 = b 2 implies that a = b ” is not true by giving a counter-example.

Answer 1

The given statement is: “For any real numbers a and b, a 2 = b 2 implies that a=b .” The given statement can be made easy by writing in the form of “if-then” as follows:

“If a and b are real numbers such that a 2 = b 2 , then a=b .”

Break the above statement into parts.

p: a and b are real numbers such that a 2 = b 2 .

r: a=b .

To show that the given statement is not true, it has to be proved that if p is true, then r is not true, i.e. it has to be proved that If a and b are real numbers such that a 2 = b 2 ,then a≠b .

Consider an example by taking a=1 and b=−1 . In this case, a≠b , while their squares are as follows:

a 2 = ( 1 ) 2 =1 b 2 = ( −1 ) 2 =1

This shows that a 2 = b 2 when a≠b .

Thus, the given statement is false.