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I am trying to write down the set of all ordered pairs $(n,m)$ such that $n=2k+1:k\in\mathbb{Z}^{+}$ and $m=2l+1:l\in\mathbb{Z}^{+}$. How would that be written in set notation?

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I would use $$\left\{(n,m)\in \mathbb{Z}^2 \mid \exists k, l \in \mathbb{Z}^+:n=2k+1,m=2l+1\right\}$$

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  • $\begingroup$ @BMehta should the $\exists$ be replaced with $\forall$ if I want all ordered pairs $(n,m)$? $\endgroup$ – Aaron Hendrickson Feb 23 '17 at 19:57
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    $\begingroup$ @AaronHendrickson No, "all" is implicit in the set notation: $\{ x \mid P(x) \}$ denotes the set of all $x$ such that $P(x)$ holds. In this particular case, you want the set of all $(n,m)$ for which there exist $k$ and $l$ with the desired property. $\endgroup$ – Uwe Feb 23 '17 at 21:04
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One way is: $$ \{ (2k+1, 2l+1) \mid k, l \in \mathbb{Z}^+ \} $$

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one another way is: $$ \{x|\exists y, z\in \Bbb{Z}: x=(y,z) \wedge \exists k, l \in \Bbb{Z}^+: y=2k+1 \wedge z=2l+1\}$$

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