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Archangel Macsika

Find a common domain for the variables x, y, z, and w for which the statement ∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z)) is true and another common domain for these variables for which it is false

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Let us find a common domain for the variables x, y, z, and w for which the statement ∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z)) is true and another common domain for these variables for which it is false.


Let the domain D contain four elements, D=\{1,2,3,4\}. Then for each elements x,y,z\in D the set D\setminus \{x,y,z\} contains at least one element, and hence we can get w\in D\setminus \{x,y,z\}.

Therefore, for this domain the statement ∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z)) is true.


Let the domain D' contain one elements, D'=\{1\}. Then for each elements x,y,z,w\in D' it follows that x=y=z=w=1, and hence there is no element w\in D' such that (w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z).

Therefore, for this domain D' the statement ∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z)) is false.

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Question ID: mtid-5-stid-8-sqid-993-qpid-848