A counterexample is any example proving that a statement is false. You only need one counterexample to prove that a statement is false.
Is each statement true or false? If the statement is false, give a counterexample.
a.
The square root of a number is always smaller than the number.
The statement is false. The square root of 0.25 is a counterexample.
√(0.25) = 0.5 and 0.5 is not smaller than 0.25
b.
If a number is an irrational number, then squaring the number will also give an irrational number.
The statement is false. The √(3) is a counterexample.
√(3) = 1.73205080757 is irrational
However, [√(3)]^{2} = √(3) × √(3) = √(3×3) = √(9) = 3 and 3 is a rational number
c.
All birds can fly
The statement is false. Penguin is a counterexample.
d.
All integers are rational numbers
The statement is true
e.
No negative number is less than its absolute.
The statement is false. -1 is a counterexample.
|-1| = 1 and -1 is less than 1
f.
If an animal lives on land, then it is a mammal.
The statement is false. A whale is a counterexample.
A whale is an aquatic mammal.
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