1. ## Statements

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

2. can someone help me with this one?

i tried:

1) the function has a upper limit but then i get lost.. don't know what f(0) = 0 means.. is it a minimum (inf)?

2) no clue.. injective function?

3) ∃λ ∈ ℝ. a = λ b ; a ≠ 0,b ≠ 0, λ ≠ 0 maybe?

3. Hi

1) says that no matter how close you are from $0$ in $\mathbb{R}^+,$ there is always a lower positive real number where the function nullifies. Example: $x\rightarrow\sin(\frac{1}{x})$

2) This is not injectivity: for instance, consider the constant function $c_1:\mathbb{R}\rightarrow\mathbb{R}$ equal to $1.$ Then for all $x,y\in\mathbb{R},\ c_1(x)=c_1(y)=1,$ so " $c_1(x)\neq 0\vee c_1(y)\neq 0$" is true. But of course $c_1$ is not injective.

Assume there are two different reals where $f$ value is $0.$ Is the formula true? Conclude.

3)
∃λ ∈ ℝ. a = λ b
This is wrong because if $a\neq 0$ and $b=0,$ that would mean a and b are not linear dependent, when they are.

You have no condition about $a$ and $b,$ so they can be the zero vector. To say $a,b$ are linearly dependent, use two scalars in your formula, or consider two cases: one with $a\neq 0$ and the other with $a=0.$