Show that the statement “For any real numbers a and b, a2 = b2 imp

1.	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» 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.

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»

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.

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