What do the following mean..
1) ∀x ∈ ℝ. (x > 0 ⇒ ∃y ∈ ℝ. 0 < y ≤ x ∧ f(y) = 0)
2) ∀x, y ∈ ℝ. (x ≠ y ⇒ f(x) ≠ 0 ∨ f(y) ≠ 0)
3)Write with symbols: "Vectors a in b are linearly dependant"
AND why is this wrong (for 3)):
∃λ ∈ ℝ. a = λ b
Hi
1) says that no matter how close you are from $\displaystyle 0$ in $\displaystyle \mathbb{R}^+,$ there is always a lower positive real number where the function nullifies. Example: $\displaystyle x\rightarrow\sin(\frac{1}{x})$
2) This is not injectivity: for instance, consider the constant function $\displaystyle c_1:\mathbb{R}\rightarrow\mathbb{R}$ equal to $\displaystyle 1.$ Then for all $\displaystyle x,y\in\mathbb{R},\ c_1(x)=c_1(y)=1,$ so "$\displaystyle c_1(x)\neq 0\vee c_1(y)\neq 0$" is true. But of course $\displaystyle c_1$ is not injective.
Assume there are two different reals where $\displaystyle f$ value is $\displaystyle 0.$ Is the formula true? Conclude.
3)
This is wrong because if $\displaystyle a\neq 0$ and $\displaystyle b=0,$ that would mean a and b are not linear dependent, when they are.∃λ ∈ ℝ. a = λ b
You have no condition about $\displaystyle a$ and $\displaystyle b,$ so they can be the zero vector. To say $\displaystyle a,b$ are linearly dependent, use two scalars in your formula, or consider two cases: one with $\displaystyle a\neq 0$ and the other with $\displaystyle a=0.$