- #1

Korisnik

- 62

- 1

## Homework Statement

Prove that ##f: \mathbb{R}\to\mathbb{R}, f(x) = x^2## is not injective.

## Homework Equations

Definition of an injection: function ##f:A\to B## is an injection if and only if ##\forall a,b \in A, f(a) = f(b) \Rightarrow a = b##.

## The Attempt at a Solution

##f: \mathbb{R}\to\mathbb{R}##

Let ##a,b \in \mathscr{D_f}##, suppose ##f(a) = f(b)##, then ##a^2=b^2##. By solving this equation we get ##a=b\ or \ a = -b##.

This is where I'm stuck. How do I prove that (and is the following what I

*have*to prove) ##\forall a,b \in \mathbb{R}, (a = b \vee a=-b) \Rightarrow (a = b) \equiv F##?

By introducing the substitutions ##A := (a =b), B:= (a = -b)##, the proposition is

*reduced*to ##(A \vee B) \Rightarrow A \equiv F##.

Now, what am I looking for in the truth table? The expression is obviously not always false, but for the ##F## to

*hold*I have to find one case where the proposition does not hold and I've proven my hypothesis. From the truth table, if ##B \equiv T, A \equiv F## we get ##T \Rightarrow F##, which is false; consequently, our problem is solved.

Is my approach correct, and is there another approach? Thank you in advance.