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Math Help - Statements

  1. #1
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    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




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  2. #2
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    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?
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  3. #3
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    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.
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