For All and There Exists

For all

A quantifier says how many cases a claim covers. "For all", also read "for every", claims something of every case in a stated set. "For all real numbers n, $n^2\ge 0$n² ≥ 0" claims that the square of every real number is at least 0, with no exception. This is the universal quantifier.

One exception destroys a for-all claim. "For all whole numbers n, n is even" is false, because 3 is odd.

The set the claim ranges over is part of the claim. "For all whole numbers n, $n^2\ge n$n² ≥ n" is true: $0^2=0$0² = 0, $1^2=1$1² = 1, and every larger whole number has $n^2>n$n² > n. "For all real numbers n, $n^2\ge n$n² ≥ n" is false: $n=\frac{1}{2}$n = 1/2 gives $n^2=\frac{1}{4}$n² = 1/4, which is less than $\frac{1}{2}$1/2.

There exists

"There exists" claims only that at least one case in the set has the property. This is the existential quantifier. "There exists a whole number n with $n^2=49$n² = 49" is true, and n = 7 is enough to show it.

The set matters here too. "There exists a whole number n with $n^2=2$n² = 2" is false, since $1^2=1$1² = 1 and $2^2=4$2² = 4 and the squares of larger whole numbers are larger still. "There exists a real number n with $n^2=2$n² = 2" is true, with $n=\sqrt{2}$n = √2.

Proved and broken in opposite ways

A for-all claim about the whole numbers covers infinitely many cases, so it is proved only by an argument that reaches every case. Checking a hundred examples is not enough. It is broken by one example that fails.

A there-exists claim works the other way. One example that works proves it. To break it you must show that no case works, which is a claim about every case and needs an argument.

So "every prime is odd" falls to the single prime 2, and "some square ends in 6" is proved by 4 × 4 = 16. "There exists a whole number n with $n^2=50$n² = 50" is broken by an argument: $7^2=49$7² = 49 and $8^2=64$8² = 64, so no whole number has a square of exactly 50.

Eight values and the line at 6. The two claims at the top are for all x, $x\le 6$x ≤ 6, and there exists x with x > 6. Lift $x_5$x₅ above the line: one value is enough to make the first claim false and the second true.

The order of quantifiers

When a statement has two quantifiers, their order changes its meaning. Over the whole numbers, "for all n, there exists m with m > n" says: whatever n you are given, you can find a larger m. It is true, because m = n + 1 works for every n.

"There exists m, for all n, m > n" says something much stronger: one single m is larger than every whole number at once. That would make m larger than itself. Any m you choose fails at n = m, since m > m is false. So the second statement is false.

In the first statement, m is chosen after n and may depend on it. In the second, m must be chosen first and then work for every n. "Every person has a mother" and "there is a person who is the mother of everyone" use the same two quantifiers in opposite orders, and only the first is true.

Each column is a whole number n. The first row gives a larger number for each n, namely n + 1, so for all n there exists a larger m. The second row tries the single number 3 against every n, and it fails from n = 3 on. Any other choice of m fails in the same way at n = m.

The usual mistakes

Proving a for-all claim with examples. $n^2\ge n$n² ≥ n holds for n = 0, 1, 2 and 3, and that says nothing about the whole numbers not tried. The argument has to cover them all.

Breaking a there-exists claim with one example. n = 3 does not have $n^2=50$n² = 50, but that leaves every other n untried. The claim is false because no whole number works, as $7^2=49$7² = 49 and $8^2=64$8² = 64 show.

Swapping the order of two quantifiers as if it changed nothing. "For all n, there exists m with m > n" is true of the whole numbers, and "there exists m, for all n, m > n" is false.

Leaving out the set. "Some n has $n^2=2$n² = 2" is true for real numbers and false for whole numbers.

Lifeguard trainees

In the application below, four trainees take four assessments. "Some trainee passed every assessment" needs one trainee with every pass, a full row in the grid of passes. "Every assessment was passed by some trainee" needs only a pass in every column, possibly from a different trainee each time, and the two answers differ.