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Hello I just started real analysis and I was wondering if the notation of the statement correct?

For all real numbers, $x$, there exists a real number $z$ and a smaller real number $y$ such that $z$ is bigger than the sum of $x$ and $y$.

(i) the statement, using only mathematical symbols, (ii) the negation of the statement, using only mathematical symbols, (iii) the contrapositive of the statement, using only mathematical symbols.

i) $\forall x∈R,\, ∃z∈R \land∃y∈R:y<z \text{ such that } z>x+y$

ii)$∃ x∈R,\, \forall z∈R \land\forall y∈R:y \ge z \text{ such that } z\ge x+y$

iii) I'm a bit confused on this one because to my knowledge a contrapositive needs a implication but there is no implication in this statement or have I got it totally wrong?

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  • $\begingroup$ I was under the assumption that "such that" is the same thing as ":", and also is part i and ii correct? $\endgroup$ Aug 7, 2014 at 13:14

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