Interpreting a statement in first order logic

[Mr Davis 97]

Homework Statement

Rewrite the following statements in symbolic form:

a) If $a$ and $b$ are real numbers with $a \ne 0$, then $ax+b=0$ has a solution.
b) If $a$ and $b$ are real numbers with $a \ne 0$, then $ax+b=0$ has a unique solution.

Homework Equations

The Attempt at a Solution

Attempts at solution:

Let $P(x,a,b)$ be the statement that $ax+b=0$ is true.
a) $\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} P(x,a,b)$
b) $\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} (P(x,a,b) \wedge \forall y (P(y,a,b) \implies y=x))$

Is that at all right? Is there an easier way? It all seems very cumbersome.

 

[fresh_42]

Homework Statement

Rewrite the following statements in symbolic form:

a) If $a$ and $b$ are real numbers with $a \ne 0$, then $ax+b=0$ has a solution.
b) If $a$ and $b$ are real numbers with $a \ne 0$, then $ax+b=0$ has a unique solution.

Homework Equations

The Attempt at a Solution

Attempts at solution:

Let $P(x,a,b)$ be the statement that $ax+b=0$ is true.
a) $\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} P(x,a,b)$
b) $\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} (P(x,a,b) \wedge \forall y (P(y,a,b) \implies y=x))$

Is that at all right? Is there an easier way? It all seems very cumbersome.

Looks good to me, although I'm no logic expert, and I would add an $\in \mathbb{R}$ to the $y$.
To make it less cumbersome a professor of mine used $\exists !$ to express uniqueness, but this isn't an official symbol. Some even used a symbol for "without loss of generality".

 

[member 587159]

Looks good to me, although I'm no logic expert, and I would add an $\in \mathbb{R}$ to the $y$.
To make it less cumbersome a professor of mine used $\exists !$ to express uniqueness, but this isn't an official symbol. Some even used a symbol for "without loss of generality".

Didn't know that this isn't an official symbol. I use it all the time.

 

[fresh_42]

Didn't know that this isn't an official symbol. I use it all the time.

I don't know for certain, I only checked it in Wikipedia. But it makes sense, as it combines two statements in one. It is convenient, but not pure in a logical sense.