1. ## laws of logic

i have just started discrete maths at uni and we have been given this

"Use the laws of logic to rewrite the following expressions and determine whether each is a tautology, contradiction or neither

a) (p or q) or (-(p and (q -> p)) or r)))
b) (((p -> r) -> p) and q) and (-((p -> r) ->p))"

we have only been told the rules and not how to use them, yet this question is in an assignment, can someone go through this as simply as possible using the names of the rules, thanks

2. For the first problem (in case you don't use this notation, $\displaystyle \lor$ means or, $\displaystyle \land$ means and, and $\displaystyle \lnot$ means not):

1. First thing is to get rid of the implication. I'm not sure what this rule is called, my prof called it implication reduction but your textbook might call it something else. At any rate,
$\displaystyle (p \to q) \Leftrightarrow (\lnot p \lor q)$

2. Now use the distributive property,
$\displaystyle p \land (\lnot p \lor q) = (p \land \lnot p) \lor (p \land q)$

3. From there, recognise that $\displaystyle (p \land \lnot p)$ is always false, and that (false $\displaystyle \lor$ x) = x (where x is any logical expression).

4. Then use DeMorgan's law to distribute the $\displaystyle \lnot$ in the second set of brackets.
$\displaystyle \lnot(p \land q) = \lnot p \lor \lnot q$

5. At this point your expression should have only $\displaystyle \lor$'s in it. You can use the associative law to remove all the brackets.

6. Notice that $\displaystyle p \lor \lnot p$ is always true, and true $\displaystyle \lor$ x = true (where x is any logical expression).

3. Originally Posted by tashworth

we have only been told the rules and not how to use them, yet this question is in an assignment, can someone go through this as simply as possible using the names of the rules, thanks
Oddly enough, I was in the same lecture theatre when you were given those rules, and I distinctly remember being shown several examples of how to use them at the same time.

The onus is then on you to practice using the tutorial and practical exercises, as well as the exercises in your textbook.

If you're still having trouble, visit the Numeracy Centre at C5A.225

4. Originally Posted by yageriswatching
Oddly enough, I was in the same lecture theatre when you were given those rules, and I distinctly remember being shown several examples of how to use them at the same time.

The onus is then on you to practice using the tutorial and practical exercises, as well as the exercises in your textbook.

If you're still having trouble, visit the Numeracy Centre at C5A.225
That's creepy man, haha.