# How does r∪(-p∩q∩-r) simplify to r∪(-p∩q) ?

• B
Gold Member
How does r∪(-p∩q∩-r) simplify to r∪(-p∩q) ? The second expression is just the first with the "-r" gone at the end. I'm not seeing how to get from the first expression to the second using any of the basic laws like distribution, de morgan, tautology, etc. What am I missing?

andrewkirk
Homework Helper
Gold Member
Let ##U## be the set of which all these sets r,p,q are subsets, so that ##-r=U-r## and ##U=r\cup -r##.
Then we have:
\begin{align*}
r\cup(-p\cap q\cap -r)
&= r\cup((-p\cap q)\cap -r) \quad\textrm{[associative law]}
\\&=
(r\cup(-p\cap q)) \cap( r\cup-r)
\\&=
(r\cup(-p\cap q)) \cap U
\\&=
r\cup(-p\cap q)
\quad\textrm{[since $U\cap A=A$ for any $A\subseteq A$]}
\end{align*}

Stephen Tashi
After you do distribution, you can represent ##(r \cup \neg r)## as "##T##" since its always true. Then ##(A) \cap T## is equivalent to ##(A)##.

Your text materials may use a different technique than the notation "T". They should have some rule equivalent to the effect of using "T".

FactChecker
So $$r \cup (-p \cap q) = r \cup ((-p \cap q \cap r) \cup (-p \cap q \cap -r))$$
$$= (r \cup (-p \cap q \cap r)) \cup (r \cup (-p \cap q \cap -r))$$
$$= r \cup (r \cup (-p \cap q \cap -r))$$
$$= r \cup (-p \cap q \cap -r)$$