Barry has taught mathematics at the college and high school level and has a master's degree in teaching secondary mathematics.
Counterexamples are a useful tool in mathematics. Learn what a counterexample is and how it can be used to prove the boundaries of theorems. You will also look at some examples across different branches of mathematics.
What Is a Counterexample?
A counterexample is a special kind of example that disproves a statement or proposition. Counterexamples are often used in math to prove the boundaries of possible theorems. In algebra, geometry, and other branches of mathematics, a theorem is a rule expressed by symbols or a formula. Counterexamples are helpful because they make it easier for mathematicians to quickly show that certain conjectures, or ideas, are false. This allows mathematicians to save time and focus their efforts on ideas to produce provable theorems.
Counterexamples are often used in math, but the truth is, counterexamples are all around us. Consider the statement, 'All mathematicians have crazy hair.' Now consider the difficulty you would encounter trying to prove this statement. Would this even be possible? It can often be very difficult to prove a proposition is true, so our efforts are better focused to find a counterexample and prove the proposition false. It only takes one counterexample to make the proposition false.
In geometry, the study of logic and proof is very important. Counterexamples are often discussed, as they can be used to disprove conditional statements. A conditional statement has two parts, such as: 'If you study algebra every night, then you will make 100 on your next test.' There are likely many examples that could be provided to make this statement true, but again, we only need one counterexample to prove it false. To find a counterexample to a conditional statement, you need an example to make the initial condition true, but at the same time, make the concluded statement false. Any student who studies algebra every night would satisfy the initial condition, and if one of these students did not make 100 on their next test, they would not satisfy the concluded statement. Thus, the first part is true, while the second is false. This student would be a counterexample.
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To consider an example from geometry, consider the proposition, 'All right triangles are scalene.' Again, it would be very time consuming to try and prove this proposition is true. There are an infinite amount of right triangles that can be drawn if all possibilities are to be considered, so any efforts to try and prove all cases would be wasted. We only need one counterexample, however, to prove this false. Consider a right triangle with two 45 degree angles; it would have two equal bases making it isosceles. This triangle would be a counterexample.
For an example from algebra, we can consider the proposition, 'All prime numbers are odd.' This one would seem difficult to disprove, as even numbers are always divisible by 2, and therefore, they are composite (not prime). A counterexample for this statement would be the number 2. The number 2 has no divisors other than itself and 1, therefore, it is prime. It is a counterexample.
Let's review. A counterexample is an example that disproves a proposition. Counterexamples exist all around us in the world and are often used in mathematics to prove propositions are false. Counterexamples are important to geometry for proving conditional statements false. This is achieved by providing an example that satisfies the initial condition but makes the concluded statement false. Counterexamples are also useful in other fields of mathematics to prove initial theories and propositions false, allowing us to focus our efforts on other possible theories. To prove a proposition is true, all cases must be proven, but it only takes one counterexample to prove a proposition is false.
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