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
1) says that no matter how close you are from in there is always a lower positive real number where the function nullifies. Example:
2) This is not injectivity: for instance, consider the constant function equal to Then for all so " " is true. But of course is not injective.
Assume there are two different reals where value is Is the formula true? Conclude.
This is wrong because if and that would mean a and b are not linear dependent, when they are.∃λ ∈ ℝ. a = λ b
You have no condition about and so they can be the zero vector. To say are linearly dependent, use two scalars in your formula, or consider two cases: one with and the other with