MATH1050 Handout: Dis-proof by counter-example
To dis-prove a statement is the same as to prove that the statement concerned is false.
Very often a statement to be dis-proved is of the form below:
• ‘Suppose so-and-so satisfy blah-blah-blah. Then so-and-so satisfies bleh-bleh-bleh.’
In such a situation, we proceed by naming one ‘concrete’ so-and-so which indeed satisfies blah-blah-blah
and which fails to satisfies bleh-bleh-bleh. Such a concrete so-and-so is called a counter-example for
the statement to be dis-proved.
Example (1). Let x, y ∈ Z. Suppose x is divisible by y and y is divisible by x. Then x = y.
Dis-proof by counter-example.
Take x = 1, y = −1. x, y ∈ Z. x is divisible by y and y is divisible by x. But x 6= y.
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Example (2). Let a, b be rational numbers. a + b 2 is irrational.
Dis-proof by counter-example.
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Take a = 0, b = 0. a, b are rational numbers. a + b 2 = 0. a + b 2 is not irrational.
Example (3). Let r, s, t ∈ R. Suppose r is a non-zero rational number and s is an irrational number.
Then both rs + t, rs − t are irrational numbers.
Dis-proof by counter-example.
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Take r = 1, s = 2, t = 2. r is a non-zero rational number. s is an irrational number. Note
that rs − t = 0. One of rs + t, rs − t, namely the latter, is not irrational.
Example (4). The product of any two distinct irrational numbers is irrational.
Remark. At first sight this statement is not of the form ‘if blah-blah-blah then bleh-bleh-bleh’. However,
it can be re-written in this way:
Let u, v be irrational numbers. Suppose u 6= v. Then uv is an irrational number.
Dis-proof by counter-example.
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Take u = 2, v = − 2. u, v are irrational numbers, and u 6= v. uv = −2, and it is not an
irrational number.
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