What is an example? What is a counter example? You are at least sure that they do counter each other very often.
George Polya and John Conway have already invested a lot of their time to help and share their views on how to build Problem Solving Skills. Do check out his book "How to Solve It?".
If you love solving problems and observing patterns in everything, then you will also find the pattern of how your mind starts the exploration, while you feed it with a problem.
The mind has a very sharp intellect. He (She) starts to explore and search the database (memory) whether there has been a similar scenario or not and cut it down into pieces like a knife just to solve your problem.
As you learn Mathematics and transcend into Advanced Mathematics, you will start to hear people saying the word "Mathematical Maturity" too often. You will get perplexed with the inferiority complex crawling into your mind, thinking that they must be really geniuses. No, they are not. You can do that too. We, are here to help you.
You ask a question. If not you must. Or if you give a definition. Whether you have a solution or not. These examples and counterexamples are the step stones to help you take steps one at a time towards the solution or understanding the definition.
Let's take an example to understand this better.
Suppose, you give the definition of a prime number.
The next step will always be showing some examples, which numbers are primes and which are not. This will help you and the students to understand why we give such a definition. Also, this will help you to explore why such a definition is valid.
Examples are powerful. You can understand it, even more, when you see that you need examples to understand why examples are important.
In our Teachers For Tomorrow Training Program, we will give such examples more. Stay Tuned.
Counterexamples are interesting. You ask a question and it may be false. The question has a solution in negative. Counterexamples are even more interesting. They are tricky to find.
Suppose, you are understanding Pythagorean Triplets i.e. Positive Integer Solutions to \( a^2 + b^2 = c^2 \). Now you take examples to understand the solutions better.
You get (3,4,5) as a solution. You try to find the next where 5 is increased a bit. You get (6,8,10).
If you are a child at heart. You may think, "Wow, are they multiples of each other always?" This is the first step. This is called the art of Generalization. Let that be a topic for a different day.
Now, you try to explore more. Then you get the numbers (5,12,13). You are sad. The question, you asked excitedly has no answer in positive. This solution has an answer giving a negative example to your question. This is what Counterexample is. But, this is an opportunity to ask another question.
Then, can I find a general formula among these numbers? And you give conjectures ( questions ) and give examples and counterexamples and make your own universe of Mathematics.
Your Questions, Your Examples, Your Counterexamples, Your Proofs - Thus you discover your World of Mathematics.
There, you find yourselves blissed-out and drunk in Mathematics for hours and days. Sometimes, you may forget to eat and sleep. Then, you know it is working. :p
Don't forget to keep a journal where it is just you and your Mathematics. 🙂