Below are some statements that I would like to know how to express symbolically:
1) x is a power of 3
2) $f : \mathbb{R} \to \mathbb{R}$ is a bijective function
3) $f : \mathbb{R} \to \mathbb{R}$ is an invertible function
Here are a few clarifications:
- 1) No exponentiation or compound statements are allowed. That would make it too simple, as shown in the incorrect answer for 1) given below.
- 2) f(x) is invertible simply means that it has an inverse.
Questions:
1) Does my answer to 1) work and if so, why? A brief but comprehensible explanation will suffice.
2) Do my answers work? And if so, can they be made more concise?
Here's what I have so far:
1) Note first that no power of 3 is negative. Now, we define the compound statement that $x$ is prime. $\exists x\in\mathbb{N} (x >1 (\exists a,b\in \mathbb{N} (x=ab) \Rightarrow (a=1\vee a=x)))$.
Next we define the compound statement that $x$ is a factor of $y$ $\exists y\in \mathbb{Z}, \exists x\in\mathbb{Z} (\exists z \in \mathbb{Z} (y=xz))$.
Now, we can use these compound statements freely. $\exists x \in\mathbb{N} \wedge (\forall y\in\mathbb{N} (y$ is prime $\wedge y$ is a factor of x) $\Rightarrow y=3) \Leftrightarrow (\exists z\in \mathbb{N} (x = yz) \Rightarrow y =3z)$.
2) $\forall y \in \mathbb{R} \exists x \in \mathbb{R} (y = f(x) \Leftrightarrow (\forall z\in \mathbb{R} (y=f(z) \Rightarrow x=z)))$.
3) $\exists y \in \mathbb{R}, \forall x\in\mathbb{R} (y=f(x) \Leftrightarrow (\forall z\in\mathbb{R} (y=f(z)\Rightarrow (z=x))))$. Note: I'm starting off with an existential quantifier because the function's range does not have to be the set of real numbers (it can be a subset of the real numbers).
hint
$1)$
$$(\exists n\in \Bbb N) \;\; : \;\; x=3^n$$
$2)$
$$(\forall y\in \Bbb R) \;\; (\exists \; ! \; x\in \Bbb R) \;\; : \;\; f(x)=y$$