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Existence Proofs in Math: Definition & Examples

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  • 0:00 Existence Proofs
  • 1:24 How to Prove Existence Proofs
  • 2:45 Another Example
  • 4:34 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we define existence theorems and existence proofs. We also explain how to go about proving existence theorems and look at examples of proofs for simple existence theorems.

Existence Proofs

Suppose you have a squirrel named Flufftail as a pet. One day, while you're talking to a friend about Flufftail, a stranger overhears you and points out that a squirrel is a rare animal to have as a pet. He then goes on to tell you that because it is so rare, he doesn't believe that Flufftail exists. To which you reply that Flufftail certainly does exist, and you can prove it! Believe it or not, this scenario has some mathematical significance. You see, in mathematics, proving that Flufftail exists would be considered an existence proof.

When a theorem states that an element, call it x, exists that satisfies a certain property, we call that theorem an existence theorem, and the proof of the theorem is called an existence proof. For example, consider these existence theorems:

  • First, there exists a real number x, such that 2x - 6 = 8.
  • Second, there exists a prime number p, such that p + 8 is also a prime number.
  • And, finally, a function f exists, such that f(x) = f ' (x).

This is just the tip of a very large iceberg. There are many more out there! Since these types of theorems show up so often in mathematics, it's really helpful to know how to perform existence proofs.

How to Prove Existence Proofs

Back to Flufftail. You told the stranger you could prove that Flufftail exists. Do you have any ideas on how to do this? Common sense is probably telling you to prove it by simply showing Flufftail to this stranger, and common sense is correct. Proving existence theorems is as simple as showing that there is an element that satisfies the theorem.

Technically, existence proofs are carried out by finding or constructing an element, x, that satisfies the theorem. Because of this, these types of proofs are also commonly called constructive proofs. Let's give it a try. Consider the first existence theorem that was previously used.

  • There exists a real number x, such that 2x - 6 = 8.

To prove that this statement is true, find or construct a number x that satisfies the equation 2x - 6 = 8. You could use trial and error to find an x that gives a true statement. You could also use algebra and actually solve for x in the equation. Let's do the latter, since it's cleaner and simpler.

existence2

By solving for x, you get x = 7. Therefore, you have found a real number that makes the equation true, because 2(7) - 6 = 8. Ta-da! You've just informally proven the theorem. Huh! That was quite easy. Let's try another one!

Another Example

Look at the second existence theorem from earlier in this lesson.

  • There exists a prime number p, such that p + 8 is also a prime number.

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