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: "Vectorsainbare linearly dependant"

AND why is this wrong (for 3)):

∃λ ∈ ℝ.a= λb

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- October 21st 2009, 10:29 AMmetlxStatements
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*

- October 23rd 2009, 09:51 AMmetlx
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? - October 23rd 2009, 10:48 AMclic-clac
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)

Quote:

∃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