Definition
28
[00R] We usually write
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”\(∀ x∈ A, P(x)\)” |
to say |
”for every \(x\) in \(A\) \(P(x)\) holds”, |
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or |
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”\(∃ x∈ A, P(x)\)” |
to say |
”there is a \(x\) in \(A\) for which \(P(x)\)” holds; |
(where \(A\) is a set); to link these writings to the previous definitions, we decide that the previous writings are abbreviations for
\begin{align*} ∀ x∈ A, P(x) \doteq & ∀ x, x∈ A⇒ P(x)\quad , \\ {} ∃ x∈ A, P(x) \doteq & ∃ x, x ∈ A ∧ P(x)\quad . \end{align*}
Note that these RHS are ”well-formed formulas”. See also the exercise [016].