Proving Uniqueness

Process (Direct Proof)

1. Assume two objects have the same property
2. Show that the two are equal

Alternate Process (Contradiction)

1. Assume there exists a $y\ne x$ and that $y$ has the same property
2. Derive a contradiction

Example

Assume $\overrightarrow{0}$ and $\overrightarrow{0^{\prime }}$ are both zero vectors. This means $\overrightarrow{0}⊞\overrightarrow{x}=\overrightarrow{x}$ for all $\overrightarrow{x}$ $\overrightarrow{0^{\prime }}+\overrightarrow{x}=\overrightarrow{x}$ for all x

• Then, $\overrightarrow{0}=\overrightarrow{0^{\prime }}⊞\overrightarrow{0}$
• Then, $\overrightarrow{0}=\overrightarrow{0}⊞\overrightarrow{0^{\prime }}$ by Vector Space axiom 2 (commutative)
• $\overrightarrow{0}=\overrightarrow{0^{\prime }}$ by hypothesis on $\overrightarrow{0}$ $\text{squarse}$