# How can I use a truth table to show that this is a tautology?

How can I show that this is a tautology by using a truth table? $(p∨q)∧(¬p∨r)\to(q∨r)$

I know how to do it by logical equivalences, but now I have to use a truth table. Never done it before so I dont even know where to start.

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The idea is that you create a truth table for the whole statement. If the "output" column is only "true" values, then it's a tautology (by definition). It helps to build this truth table slowly, adding only one binary operation at a time:

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

I've left out some rows so there's still some fun for you to have... ;)

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$\TeX$ing, like a boss. – Pedro Tamaroff May 2 '13 at 22:44

To make a truth table, you make columns for all the variables and rows for all combinations of truth values of the variables. Then you make as many columns as you want to assess the truth value of the statement in question. If you want to prove something a tautology, it must be true for all values of the truth value of the variables. I'll give some lines of the table:

$$\begin {array}{c|c|c|c|c|c} \\ p&q&r&p \vee q&\lnot p \vee r&q \vee r \\ \hline T&T&T&T&T&T\\T&T&F&T&T&T\\F&F&F&F&T&F \end {array}$$

There are five more lines. Mine would usually be the first, second, and eighth lines in the table. Now add some columns to build up your expression.

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Thanks to both of you! It really helped! – Maren May 2 '13 at 22:24

"I don't know where to start." Then read the appropriate chapter(s) of some good textbook to get the principle of using truth-tables before asking here (for it is the general technique you need to know).

There are a lot of good books to choose from! I happen to like P*t*r Sm*th's Introduction to Formal Logic. Among freely available treatments, Paul Teller's Modern Formal Logic Primer is now available from his website and is very good.

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