Think about the following sentence.
All poodles are dogs, but not all dogs are poodles.
Why do you think this statement is true?
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This statement is true. Although poodles are dogs, other dogs include collies, beagles, and chihuahuas.
Is the following statement true or false? How do you know?
If you have a dog, then it is a poodle.
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The statement is not true. A counterexample would be you have a dog, but it is a bulldog.
Counterexamples are used in mathematics to disprove statements that are false. They usually point to a case that shows the statement is not always true.
To hear a song about counter examples, watch the following video.
Source: Algebra man counter example song, givinituptooquick, YouTube