Uniqueness Proof procedure

Question

I'm reading a book on understanding math proofs to enable me to understand mathematics at a deeper level. Along the way I came across this:

An element belonging to some prescribed set $A$ and possessing a certain property $P$ is unique if it is the only element of $A$ having property $P$. Typically, to prove that only one element of $A$ has property $P$, we proceed in one of two ways:

(1) We assume that $a$ and $b$ are elements of $A$ possessing property $P$ and show that $a = b$.

<p>(2) We assume that $a$ and $b$ are distinct elements of $A$ possessing property $P$ and show that $a = b$.</p>

The problem is points (1) and (2) sound exactly the same to me. The only clue that they are different is in the word "distinct", which the author did not bother to explain what it meant in this context.

If $a$ and $b$ are distinct, aren't they suppose to mean $a≠b$ ? So, how can we later show that $a=b$?

Answer 1

You are correct in that they share the same proof method, but they are different in their approach. (1) is a direct proof, while (2) is a proof by contradiction.

Let's take an example: proving that $1$ is the unique element of the set $A=\{1,2,4,6\}$ with the property that $x^3=1$.

Method (1): Let $a,b\in A$ such that $a^3=1$ and $b^3=1$. Now we note that every element of $A$ is a real number, and the only solution to the equation $x^3=1$ over the reals is $x=1$. Hence $a=b=1$ (we need to also note that $1\in A$).

Method (2): Assume $a,b\in A$ such that $a^3=1$, $b^3=1$, and $a\ne b$. But $a^3=b^3$ which implies (over the reals) that $a=b$. This is a contradiction, so if there is an element of $A$ which satisfies $x^3=1$, then it is unique.