# Calculating Truth Tables for Propositions

• MHB
• evinda
In summary, the conversation discusses calculating truth tables for two propositions: $$(p \land q) \lor (\lnot p \land q) \to q$$ and $$p \land \lnot q \to r$$. The tables show that the first proposition is always true, while the second proposition is only true when $p=q=1$ or $p=0, q=1$. The conversation confirms that the calculations are correct.
evinda
Gold Member
MHB
Hello! (Wave)

I want to calculate the truth tables of the following propositions:

$$(p \land q) \lor (\lnot p \land q) \to q \\ p \land \lnot q \to r$$

I have done the following:\begin{equation*}
\begin{array}{c|c|c|c|c}
p & q & p \land q & \lnot p \land q & (p \land q) \lor (\lnot p \land q) \to q \\
\hline
1 & 1 & 1 & 0 & 1 \\
1 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 & 1
\end{array}
\end{equation*}

and

\begin{equation*}
\begin{array}{c|c|c|c}
p & q & \lnot q & p \land \lnot q \\
\hline
1 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0
\end{array}
\end{equation*}

If $p=q=1$ and $r=0$, then $p \land \lnot q \to r$ is true, and the same holds if $r=1$. The same holds when $p=q=0$ and $p=0$, $q=1$.

If $p=1$ and $q=0$, then if $r=0$ then $p \land \lnot q \to r$ is false, and if $r=1$ then it is true.
Is everything right? Or have I done something wrong? :unsure:

Hi evinda,

That looks correct

castor28 said:
Hi evinda,

That looks correct

Nice, thank you! :)

## What is a truth table?

A truth table is a table that shows all the possible combinations of truth values for a given set of propositions and the resulting truth value for a compound proposition.

## How do you calculate a truth table?

To calculate a truth table, you first list all the possible combinations of truth values for the individual propositions. Then, using logical operators such as AND, OR, and NOT, you determine the resulting truth value for the compound proposition for each combination of truth values.

## Why is calculating truth tables important?

Calculating truth tables is important because it allows us to determine the truth value of complex propositions based on the truth values of their individual components. It also helps us to identify logical relationships and evaluate arguments for validity.

## What are the basic logical operators used in calculating truth tables?

The basic logical operators used in calculating truth tables are AND (represented by ∧), OR (represented by ∨), and NOT (represented by ¬). These operators are used to combine propositions and determine the resulting truth value.

## Can truth tables be used for propositions with more than two components?

Yes, truth tables can be used for propositions with more than two components. In these cases, the number of rows in the truth table will increase exponentially based on the number of components. However, the basic principles of calculating truth tables remain the same.

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