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Counterexamples are simply the such examples for a given statement ,which meets it's requirement or all conditions to obtain a certain result, yet doesnot produce the same result, thus it contradicts its conclusion but at the same time satisfying it's all necessary conditions . The best example of it is Money makes a man happy And it's counterexample is : Ram remains sad despite of rich lifestyle So , the above example , fulfills the condition of being having lot of money by Ram but it contradicts it's result of being happy , as Ram is sad despite of all money ,. May be Ram has recently lost some dear or near one which made him sad about it
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Counter example is simply the converse example of the statement.

For Example:

Every number ending with 0,2,4,6,8 are even numbers.

Counter Example: Every number which is not ending with 0,2,4,6,8 are not even numbers.

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An example that disproves a statement (shows that it is false). Example: the statement "all dogs are hairy" can be proved false by finding just one hairless dog (the counterexample).

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If you give an example which disproves the statement (the opposite person says) is called COUNTER.

EXAMPLE: All mathematics students are good at Physics but if you give anyone's example who is the "topper" of mathematics but not good at physics.

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