Let \(P(x, y)\) be a propositional function, then .................. is a tautology.
\(\exists x \forall y P(x, y) \rightarrow \forall y \exists x P(x, y)\)
Claim: \(\exists x \forall y P(x, y) \rightarrow \forall y \exists x P(x, y)\) is always true.
Assume the left side: some \(a\) has \(P(a, y)\) for every \(y\). Then for any \(y\), \(x = a\) gives \(P(x, y)\), so \(\forall y \exists x P(x, y)\) holds.
Counterexample for \(\exists x \forall y P(x, y) \rightarrow \forall y P(x0, y)\): take \(P(x, y): x = 0\) on integers and \(x0 = 1\); the left side is true, the right side false.
Counterexample for \(\exists x P(x, y0) \rightarrow \exists x \forall y P(x, y)\): \(P(x, y): x = y\); \(x = y0\) works but no single \(x\) equals every \(y\).
Counterexample for \(\exists x P(x, y0) \rightarrow \forall y P(x0, y)\): \(P(x, y): x = y\) again makes the left true and the right false.
Hence, \(\exists x \forall y P(x, y) \rightarrow \forall y \exists x P(x, y)\).
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.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
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(i) Kernel
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(iii) Nucleus
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Codes:
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(i) Machine language
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Codes:
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Which part of the QR code represents the encoded data?
