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Is this notation correct? (Learning basics of logical notation)

∃x∈X: P(x)
∀yP(y) -> O(y)
∴∃x∈X: O(x)

"There exists an element x in a set X with property P." "For all elements with property P has also property O." "Therefore there exists an element in the set X with property O."

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    Better : "All elements with property P have also property O." Parentheses are useful in order to correctly define the *scope* of quantifiers : $\forall y (Py \to Oy)$. – 2017-01-03

1 Answers 1

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Looks pretty good, but for this site you can use MathJax rather than Unicode special characters. For this specific notation, use:

$$\exists x \in X : P(x)$$
$$\forall y (P(y) \Rightarrow O(y))$$
$$\therefore \exists x \in X : O(x)$$

which produces:

$$\exists x \in X : P(x)$$ $$\forall y (P(y) \Rightarrow O(y))$$ $$\therefore \exists x \in X : O(x)$$

As for reading this out loud, I would verbalize it as follows:

$$\exists x \in X : P(x)$$ There exists an element $x$ in $X$ for which predicate $P$ of $x$ is true. $$\forall y (P(y) \Rightarrow O(y))$$ For all $y$, $P$ of $y$ implies $O$ of $y$. (Or if I want to be explicit, "For all $y$, predicate $P$ of $y$ implies predicate $O$ of $y$.") $$\therefore \exists x \in X : O(x)$$ Therefore, there exists an element $x$ in $X$ for which predicate $O$ of $x$ holds true.