[Math] Writing an expression using logic

discrete mathematicslogicpropositional-calculus

Write an expression using letters $\land, \lor, and$ $\neg$ which has the following truth table:

$$\begin{array}{ccc|c}
P&Q&R&???\\ \hline
T&T&T&F\\
T&T&F&T\\
T&F&T&F\\
T&F&F&T\\
F&T&T&T\\
F&T&F&F\\
F&F&T&F\\
F&F&F&F
\end{array}
$$

How would you in general solve for ??? I was told there is a mechanical way of doing it but I don't see it.

Best Answer

I’ll get you started. Look at the last column of the truth table: you want your statement to be true when

$$P\text{ is true and }Q\text{ is true and }R\text{ is false}\tag{1}$$

$$\mathbf{OR}$$

$$P\text{ is true and }Q\text{ is false and }R\text{ is false}\tag{2}$$

$$\mathbf{OR}$$

$$P\text{ is false and }Q\text{ is true and }R\text{ is true}\;,\tag{3}$$

and for no other combination of truth values of $P,Q$, and $R$.

Is there a simple statement that’s true exactly when $P$ and $Q$ are true and $R$ is false? Sure: $$P\land Q\land\neg R\;.$$

Now find a similar statement that is true exactly when $(2)$ is true, and another one that’s true exactly when $(3)$ is true. Once you have these three statements, how should you combine them to get what you want?

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