# There exists

If is a predicate depending only on , then "there exists such that P" (written or ) is a proposition. It is true if there is some value that makes evaluate to true.

is sometimes called the existential quantifier.

• To prove that there exists an such that holds, it suffices to give a specific and then prove that is true for that . Such a proof usually starts "let ", and then goes on to prove that holds for the given . is sometimes referred to as a witness for .
• If you know there exists some satisfying , you can use it in a proof by treating as an arbitrary value. is arbitrary because the only thing you know about is that it exists, not what its value is.

When trying to prove an existential statement , you need to give a specific value of (a witness).

Often, in a proof, it is not immediately obvious what the witness should be. Finding one often involves solving some equations or combining some known values.

One nice technique for finding a witness is to simply leave a blank space for the value of and continue on with your proof of . As you go, you may need to satisfy certain properties (for example, maybe you need at one point, and later you need ). You can make a "wishlist" on the side of your proof, reminding you of all the properties you want to satisfy. Once you've completed your proof, you can go back and find a specific value of (say, ) that satisfies all of your wishes.