Definition:Propositional Function/Examples

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Examples

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

Let $P \left({x}\right)$ be the propositional function defined as:

$x$ is even

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

\(\displaystyle P \left({1}\right)\) \(=\) \(\displaystyle F\) $\quad$ $\quad$
\(\displaystyle P \left({2}\right)\) \(=\) \(\displaystyle T\) $\quad$ $\quad$
\(\displaystyle P \left({591}\right)\) \(=\) \(\displaystyle F\) $\quad$ $\quad$

Thus $P \left({x}\right)$ is a unary propositional function (pronounced yoo-nary).

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Let $P \left({x, y}\right)$ be the propositional function defined as:

$x$ is less than $y$

Then we can create the propositional statements:

\(\displaystyle P \left({1, 2}\right)\) \(=\) \(\displaystyle T\) $\quad$ $\quad$
\(\displaystyle P \left({2, 1}\right)\) \(=\) \(\displaystyle F\) $\quad$ $\quad$
\(\displaystyle P \left({3, 3}\right)\) \(=\) \(\displaystyle F\) $\quad$ $\quad$

Thus $P \left({x, y}\right)$ is a binary propositional function .

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Let $P \left({x, y, z}\right)$ be the propositional function defined as:

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

Then:

\(\displaystyle P \left({1, 2, 3}\right)\) \(=\) \(\displaystyle F\) $\quad$ $\quad$
\(\displaystyle P \left({2, 1, 3}\right)\) \(=\) \(\displaystyle T\) $\quad$ $\quad$
\(\displaystyle P \left({5, 4, 3}\right)\) \(=\) \(\displaystyle F\) $\quad$ $\quad$

Thus $P \left({x, y, z}\right)$ is a ternary propositional function .

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