Uniqueness Proofs

What is a Uniqueness Proof?

Claim: “There is exactly one x such that…”

How to prove it: Two steps.

Find one. Then prove no other can exist.

The Technique

For the uniqueness step:

1. Assume there are two: $a$ and $b$
2. Use the conditions to show $a = b$
3. Conclude: they’re the same, so there’s only one

“Two” that are identical = one.

Example 1: Even Prime

Claim: There is exactly one even prime number.

Step 1: Existence

• 2 is even and prime ✓

Step 2: Uniqueness

• Take any even number other than 2
• It’s divisible by 2
• So it has 2 as a factor (besides 1 and itself)
• So it’s not prime

Only 2 is both even and prime.

Every other even number fails the prime test.

Example 2: Solving an Equation

Claim: The equation $2x + 6 = 0$ has exactly one solution.

Step 1: Existence

• Try $x = -3$
• $2(-3) + 6 = -6 + 6 = 0$ ✓

A solution exists.

Step 2: Uniqueness

• Assume $a$ and $b$ are both solutions
• $2a + 6 = 0 \Rightarrow a = -3$
• $2b + 6 = 0 \Rightarrow b = -3$
• So $a = b$

There’s only one solution.

Assume two exist. Show they’re the same.

Example 3: Unique Solution

Claim: There is exactly one integer $n$ such that $n + 5 = 8$.

Step 1: Existence

• Try $n = 3$
• $3 + 5 = 8$ ✓

Step 2: Uniqueness

• Assume $a$ and $b$ are both solutions
• $a + 5 = 8 \Rightarrow a = 3$
• $b + 5 = 8 \Rightarrow b = 3$
• So $a = b$

Only one solution: 3.

Any two solutions turn out to be the same.

Summary