Thread: 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

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?

Hi

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.

3)

∃λ ∈ ℝ. a = λ b

This is wrong because if and that would mean a and b are not linear dependent, when they are.

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