Definition:Propositional Function/Examples

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Examples

Let the universe be the set of integers $\Z$.

Let $\map P x$ be the propositional function defined as:

$x$ is even

Then we can insert particular values of $x \in \Z$, for example, as follows:

\(\displaystyle \map P 1\) \(=\) \(\displaystyle \F\)
\(\displaystyle \map P 2\) \(=\) \(\displaystyle \T\)
\(\displaystyle \map P {591}\) \(=\) \(\displaystyle \F\)

Thus $\map P x$ is a unary propositional function (pronounced yoo-nary).

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Let $\map P {x, y}$ be the propositional function defined as:

$x$ is less than $y$

Then we can create the propositional statements:

\(\displaystyle \map P {1, 2}\) \(=\) \(\displaystyle \T\)
\(\displaystyle \map P {2, 1}\) \(=\) \(\displaystyle \F\)
\(\displaystyle \map P {3, 3}\) \(=\) \(\displaystyle \F\)

Thus $\map P {x, y}$ is a binary propositional function .

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Let $\map P {x, y, z}$ be the propositional function defined as:

$x$ is between $y$ and $z$.

Then:

\(\displaystyle \map P {1, 2, 3}\) \(=\) \(\displaystyle \F\)
\(\displaystyle \map P {2, 1, 3}\) \(=\) \(\displaystyle \T\)
\(\displaystyle \map P {5, 4, 3}\) \(=\) \(\displaystyle \F\)

Thus $\map P {x, y, z}$ is a ternary propositional function .

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