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I'm a student and I'm struggling to understand the basics of discrete maths. If such a noob question wont offend you, please help me understand the equation and how to prepare a truth table for this?

$$p \rightarrow (\neg q \lor \neg r) \land \neg p \land q$$

If you use any substitution or such, explain how you derived them too .

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You seem to be misunderstanding something. That's a question of logic rather than discrete mathematics. In addition you should clarify the statement --- does $p$ imply everything on the RHS or only $\lnot q \lor \lnot r$? – kahen Oct 29 '10 at 11:18
Yes, I think you are missing a few parentheses. (Note that I merely translated your original statement to $\TeX$, but you still need to clarify). – J. M. Oct 29 '10 at 11:20
Actually this is a question I got for an assignment and I haven't changed anything.. I just copied it as it appeared in the Question Paper. And I have no idea how to explain anything about it. I have given parenthesis where it is given in the question. – Threecoins Oct 31 '10 at 12:12
up vote 6 down vote accepted

This answer is meant to elaborate on yunone's earlier answer. I also assume an order of precedence so that "IF a THEN b" takes lower precedence than "a AND b" or "a OR b" — that is, I assume that the expression you must evaluate is p → ((¬q ∨ ¬r) ∧ ¬p ∧ q).

Draw out the following table (drawn below in monospaced unicode): spaces in between columns are just to indicate separation because ASCII double-lines look ugly.

 p | q | r    ¬q | ¬r | ¬q ∨ ¬r    ¬p | ¬p ∧ q | (¬q ∨ ¬r) ∧ ¬p ∧ q    p → ((¬q ∨ ¬r) ∧ ¬p ∧ q)
===|===|===  ====|====|=========  ====|========|====================  ==========================
 F | F | F       |    |               |        |
 F | F | T       |    |               |        |
 F | T | F       |    |               |        |
 F | T | T       |    |               |        |
 T | F | F       |    |               |        |
 T | F | T       |    |               |        |
 T | T | F       |    |               |        |
 T | T | T       |    |               |        |

The columns pre-filled with T and F describe, in each row, a different possible setting of your variables p, q, and r. From these, you can evaluate other expressions such as ¬q and ¬r, use these to evaluate other expressions such as (¬q ∨ ¬r), and so forth.

If it helps, introduce auxiliary variables for expressions that you have to evaluate along the way. For instance, you can define A = ¬q ∨ ¬r  and  B =  ¬p ∧ q , which will help you to evaluate the expression A ∧ B  =  (¬q ∨ ¬r) ∧ ¬p ∧ q .

Generally, just break down the expression into pieces that you can manage, and re-assemble the pieces to get the whole. (Which is a high-level description of mathematical problem solving in general, of course)

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When first doing truth tables, it's best to break a sentence into its constituent parts. Here the propositions are $p,q,r$, so have the first three columns of your truth table be possible truth values for these. Then the remaining columns should be things such as $\neg q$, $\neg r$, $\neg q \vee\neg r$, $\neg p\wedge q$, $p\to (\neg q\vee\neg r)$, finally building up to $p\to (\neg q\vee\neg r)\wedge\neg p\wedge q$.

To get started, the possible triples for $(p,q,r)$ are $(T,T,T),(T,T,F),(T,F,T),(F,T,T),(T,F,F),(F,T,F),(F,F,T),(F,F,F)$, and the rest of the table should follow easily from there. and the sentence should be easier to understand in retrospect, now that you see how each part affects each other part.

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