Question

Show that the statement “For any real numbers a and b, a² = b² implies that a = b” is not true by giving a counter-example.

Answer 1

The given statement can be written in the form of “if-then” as follows.

If a and b are real numbers such that a² = b², then a = b.

Let p: a and b are real numbers such that a² = b².

q: a = b

The given statement has to be proved false. For this purpose, it has to be proved that if p, then ∼q. To show this, two real numbers, a and b, with a² = b² are required such that a ≠ b.

Let a = 1 and b = –1

a² = (1)² = 1 and b² = (– 1)² = 1

∴ a² = b²

However, a ≠ b

Thus, it can be concluded that the given statement is false.