1
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.
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.
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Solution
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.
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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