By Madhav Kaushish
A counterexample is an example that refutes a claim. Take the statements in (a) and (b): a. All humans have their heart on the left. b. Most humans have their heart on the left.
If we find even one human whose heart is not on the left, we have proven statement (a) to be false since it says, ‘All humans have their heart on the left.’ If we do find such an example, we have the choice of rejecting (a), or refining it to something like (b). In fact, examples of people with their heart on the right side of their bodies have been documented.
Notice, there is no single example that will disprove statement (b). Why is that the case? Well, if we restate the claim to say, “99% of humans have their heart on the left,” the one example we have found might be part of the 1% of humans who do not have their heart on the left.
Statements (c) and (d) are similar to (a) and (b): c. The model predicts that Candidate A will win the election. d. The model predicts that there is a 70% chance that Candidate A will win the election.
The model in statement (c) would be a bad model if Candidate A did not win the election, since it is a categorical statement and does not admit any chance of being wrong. However, if (d) were the model we were evaluating and Candidate A ended up losing the election, you cannot instantly conclude that the model was bad.
Now take statements (e) and (f): e. In Euclidean Geometry, given a triangle, there exists a circle which circumscribes it. f. All adult human beings have exactly one heart.
In order to disprove (e) and (f), we need exactly one counterexample. However, there are no counterexamples for either of them. In the case of (e), we can actually prove the statement to be true from the axioms of Euclidean Geometry and a certain definition of circumscription. The nature of mathematical proofs is discussed in a separate document.
In the case of (f), an adult human being with more or less than one heart has never been found. While this doesn’t rule out the possibility, we assume the statement to be true till we find a counterexample.
Is a counterexample sufficient to disprove the following statements? If yes, what does such a counter-example look like? 1. All 18 year olds are taller than all 12 year olds. 2. In every village in India, the average height of 18 year olds is more than the average height of 12 year olds. 3. 18 year olds tend to be taller than 12 year olds.
I will leave these to you to think through.
Now that we have seen what sorts of statements are amenable to counter-examples, let us move on to ways of finding counter-examples.
Let us take 1 from above: All 18 year olds are taller than all 12 year olds. To find counter-examples, it is often easier if we rewrite the statement in the form ‘if…then.’ Here is what it would look like: If X is an 18 year old and Y is a 12 year old, then X is taller than Y.
So, in this case a counter example would be a pair of an 18 year old and a 12 year old such than the 12 year old is taller than the 18 year old. Notice that the example we pick fits the part after the ‘If’ and before the ‘then’ and does not fit the part after the ‘then’.
Let us take another example: The area of a rectangle is the square of any of its sides. We can rewrite this as: If X is a rectangle, then its area is the square of any of its sides.
Notice that this statement is true for a whole class of rectangles, namely squares. However, the claim above is made about all rectangles and not just some rectangles. A counter-example will be a rectangle such that its area is not the square of one of its sides. In other words, any non-square rectangle.
Now for some exercises. What do counter-examples to these nonsense statements look like? I will give you the answer to A, the easiest one! A. If X is a blook then X is a treep B. If X is a blook and X is a sweeb, then X is a treep C. If X is a blook and X is a sweeb, then X is a treep and X is a gleek D. If X is a blook or X is a sweeb, then X is a treep E. If X is a blook and X is a sweeb, then X is a treep or X is a gleek F. If X is a blook or X is a sweeb, then X is a treep or X is a gleek
A counter-example to A would be a blook which is not a treep. What would counter-examples to the rest look like?