# Truth Table/Examples

## Examples of Truth Tables

### Example: $p \implies \paren {q \lor r}$

The truth table for the WFF of propositional logic:

$p \implies \paren {q \lor r}$:

can be depicted as:

$\begin{array}{c|c|c|c|c} p & q & r & q \lor r & p \implies \paren {q \lor r} \\ \hline \F & \F & \F & \F & \T \\ \F & \F & \T & \T & \T \\ \F & \T & \F & \T & \T \\ \F & \T & \T & \T & \T \\ \T & \F & \F & \F & \F \\ \T & \F & \T & \T & \T \\ \T & \T & \F & \T & \T \\ \T & \T & \T & \T & \T \\ \end{array}$

### Example: $\paren {\lnot p} \land \paren {\lnot q}$

The truth table for the WFF of propositional logic:

$\paren {\lnot p} \land \paren {\lnot q}$:

can be depicted as:

$\begin{array}{cc|c|cc} (\lnot & p) & \land & (\lnot & q) \\ \hline \T & \F & \T & \T & \F \\ \T & \F & \F & \F & \T \\ \F & \T & \F & \T & \F \\ \F & \T & \F & \F & \T \\ \end{array}$

### Example: $\lnot \paren {\paren {p \implies q} \implies \paren {\lnot \paren {q \implies p} } }$

The truth table for the WFF of propositional logic:

$\lnot \paren {\paren {p \implies q} \implies \paren {\lnot \paren {q \implies p} } }$:

can be depicted as:

$\begin{array}{c|ccc|c|cccc} \lnot & ((p & \implies & q) & \implies & (\lnot & (q & \implies & p))) \\ \hline \T & \F & \T & \F & \F & \F & \F & \T & \F \\ \F & \F & \T & \T & \T & \T & \T & \F & \F \\ \F & \T & \F & \F & \T & \F & \F & \T & \T \\ \T & \T & \T & \T & \F & \F & \T & \T & \T \\ \end{array}$

### Example: $p \implies \paren {q \implies r}$

The truth table for the WFF of propositional logic:

$p \implies \paren {q \implies r}$:

can be depicted as:

$\begin{array}{c|c|ccc} p & \implies & (q & \implies & r) \\ \hline \F & \T & \F & \T & \F \\ \F & \T & \F & \T & \T \\ \F & \T & \T & \F & \F \\ \F & \T & \T & \T & \T \\ \T & \T & \F & \T & \F \\ \T & \T & \F & \T & \T \\ \T & \F & \T & \F & \F \\ \T & \T & \T & \T & \T \\ \end{array}$

### Example: $\paren {p \land q} \implies r$

The truth table for the WFF of propositional logic:

$\paren {p \land q} \implies r$:

can be depicted as:

$\begin{array}{ccc|c|c} (p & \land & q) & \implies & r \\ \hline \F & \F & \F & \T & \F \\ \F & \F & \F & \T & \T \\ \F & \F & \T & \T & \F \\ \F & \F & \T & \T & \T \\ \T & \F & \F & \T & \F \\ \T & \F & \F & \T & \T \\ \T & \T & \T & \F & \F \\ \T & \T & \T & \T & \T \\ \end{array}$

### Example: $\paren {p \iff \paren {\lnot q} } \lor q$

The truth table for the WFF of propositional logic:

$\paren {p \iff \paren {\lnot q} } \lor q$:

can be depicted as:

$\begin{array}{cccc|c|c} (p & \iff & (\lnot & q)) & \lor & q \\ \hline \F & \F & \T & \F & \F & \F \\ \F & \T & \F & \T & \T & \T \\ \T & \T & \T & \F & \T & \F \\ \T & \F & \F & \T & \T & \T \\ \end{array}$

### Example: $\paren {p \land q} \lor \paren {r \land s}$

The truth table for the WFF of propositional logic:

$\paren {p \land q} \lor \paren {r \land s}$:

can be depicted as:

$\begin{array}{ccc|c|ccc} (p & \land & q) & \lor & (r & \land & s) \\ \hline \F & \F & \F & \F & \F & \F & \F \\ \F & \F & \F & \F & \F & \F & \T \\ \F & \F & \F & \F & \T & \F & \F \\ \F & \F & \F & \T & \T & \T & \T \\ \F & \F & \T & \F & \F & \F & \F \\ \F & \F & \T & \F & \F & \F & \T \\ \F & \F & \T & \F & \T & \F & \F \\ \F & \F & \T & \T & \T & \T & \T \\ \T & \F & \F & \F & \F & \F & \F \\ \T & \F & \F & \F & \F & \F & \T \\ \T & \F & \F & \F & \T & \F & \F \\ \T & \F & \F & \T & \T & \T & \T \\ \T & \T & \T & \T & \F & \F & \F \\ \T & \T & \T & \T & \F & \F & \T \\ \T & \T & \T & \T & \T & \F & \F \\ \T & \T & \T & \T & \T & \T & \T \\ \end{array}$

### Example: $\paren {\paren {\lnot p} \land q} \implies \paren {\paren {\lnot q} \land r}$

The truth table for the WFF of propositional logic:

$\paren {\paren {\lnot p} \land q} \implies \paren {\paren {\lnot q} \land r}$:

can be depicted as:

$\begin{array}{|cccc|c|cccc|} \hline ((\lnot & p) & \land & q) & \implies & ((\lnot & q) & \land & r) \\ \hline \T & \F & \F & \F & \T & \T & \F & \F & \F \\ \T & \F & \F & \F & \T & \T & \F & \T & \T \\ \T & \F & \T & \T & \F & \F & \T & \F & \F \\ \T & \F & \T & \T & \T & \F & \T & \T & \T \\ \F & \T & \F & \F & \T & \T & \F & \T & \F \\ \F & \T & \F & \F & \T & \T & \F & \T & \T \\ \F & \T & \F & \T & \T & \F & \T & \F & \F \\ \F & \T & \F & \T & \T & \F & \T & \F & \T \\ \hline \end{array}$