# GATE1988-12iic

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Using Armstrong’s axioms of functional dependency derive the following rules:

$\{ x \rightarrow y, \: z \subset y \} \mid= x \rightarrow z$

(Note: $x \rightarrow y$ denotes $y$ is functionally dependent on $x$, $z \subseteq y$ denotes $z$ is subset of $y$, and $\mid =$ means derives).

$\because z\subset y$, Trivially $y\rightarrow z$ . Now by transitivity, $x \rightarrow y, y \rightarrow z \implies x \rightarrow z$

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According to Wikipedia:

There is a proper subset. And here it is just a subset. Why doesn't it make a difference?

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Because proper subset is also a subset with a stricter version.

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Using Armstrong’s axioms of functional dependency derive the following rules: $\{ x \rightarrow y, \: wy \rightarrow z \} \mid= xw \rightarrow z$ (Note: $x \rightarrow y$ denotes $y$ is functionally dependent on $x$, $z \subseteq y$ denotes $z$ is subset of $y$, and $\mid =$ means derives).
Using Armstrong’s axioms of functional dependency derive the following rules: $\{ x \rightarrow y, \: x \rightarrow z \} \mid= x \rightarrow yz$ (Note: $x \rightarrow y$ denotes $y$ is functionally dependent on $x$, $z \subseteq y$ denotes $z$ is subset of $y$, and $\mid =$ means derives).