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Let's say we have a cumulative distribution function (cdf) G and random numbers v1 and v2.

The definition of strict increasing function is: v1 < v2 => G(v1) < G(v2).

In a statistics book, the author writes:

"...but with the additional assumption that the cdf G is a strictly increasing function. That is, v1 < v2 <=> G(v1) < G(v2)".

a) He writes the definition with an equivalence (<=>) and not an implication (=>). Could someone explain why? Does in fact the definition also imply that its is an equivalence?

b) The authors definition: "v1 < v2 <=> G(v1) < G(v2)" must imply that:

v1 = v2 <=> G(v1) = G(v2). Correct?

Any help is very appreciated!

The definition of strict increasing function is: v1 < v2 => G(v1) < G(v2).

In a statistics book, the author writes:

"...but with the additional assumption that the cdf G is a strictly increasing function. That is, v1 < v2 <=> G(v1) < G(v2)".

a) He writes the definition with an equivalence (<=>) and not an implication (=>). Could someone explain why? Does in fact the definition also imply that its is an equivalence?

b) The authors definition: "v1 < v2 <=> G(v1) < G(v2)" must imply that:

v1 = v2 <=> G(v1) = G(v2). Correct?

Any help is very appreciated!

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