Discrete mathematics. I solved the last two problems, but I am having a lot of trouble with the first one. I cannot reduce the proposition. What do I do?

Also, if any of the others are incorrect, feel free correct my mistakes.

Thank you for your hospitality and help.

Home - > Students B 1-2 Tool Access: Introduction to ( x 6 Module One Problem Set - Onlin X C D overleaf.com/project/5f436f400602f60001ce4c09 9 G lli Apps M Gmail YouTube Maps Translate 6 Menu Module One Problem Set Ab Review Share Submit History Chat Source Rich Text Recompile main.tex A 165 item $M(x) : \\; x$ had migraines!! !\\ PROBLEM 2 166 snhu_logo.png 167 \\end{itemize} Use De Morgan's law for quantified statements and the laws of propositional 168 Translate each of the following statements into a logical expression. Then negate the expression by adding logic to show the following equivalences: a negation operation to the beginning of the expression. Apply De Morgan's law until each negation operation applies directly to a predicate and then translate the logical expression back into English. !Ill (a) -VI (P(z) A-Q(1)) = 1 (-P(1) VQ(z)) 169 170 flit (b) VI (-P(I) + Q(1)) = 1: (-P(I) A-Q(I)) A 171 Sample question: Some patient was given the placebo and the medication. !\\ 172 + \\begin{itemize} Given ( P(x) Q(x)) (P(x) Q(x)) Take L.H.S (P(x) Q(x)) A 173 \\item S\\exists x\\; (P(x) \\; \\land \\; D(x))$\\\\ (P(x))V Q(x))LawofImplies(P) - P V Q 4 174 \\item Negation: $\ eg \\exists x\\; (P(x)\\; \\land \\; D(x))$\\\\ (P(x) V Q(x))ByDeMorgan0 175 item Applying De Morgan's law: $\\forall x\\; (\ eg P(x) \\; \\lor \\; \ eg D(x))$\\\\ slaw(P) - P A 176 \\item English: Every patient was either not given the placebo or not given the medication (or both) . !\\ (P(x)V Q(x))ByDeMorgan0 slaw() - X 177 \\end{itemize} (P(x) Q(x))ByDeMorgan0 178 slaw(P V Q) - Pmedia 179 \ ewpage (P(x) Q(x)) 180 R.H.S L.H.S R.H.S 181 182 - \\begin{enumerate} [label=(\\alph*) ] 183 (c) -3x ( P(I) V (Q(I) A -R(I)) ) = VI(P(I) A(-Q(I) V R(I))) A 184 \\item Every patient was given the medication or the placebo or both. !! !\\ 185 *Enter your answer below this comment line. 186 \\\\\\\\ Logical Expression: -(x) (P(x) V D(x)) (P(x) V (Q(x) R(x))) (P(x) (Q(x) V R(x))) Take L.H.S (P(x) V (Q(x) R(x))) 187 Negation : (x) (P(x) V D(x)) (x) ( P(x) D(x)) (P(x) V (Q(x) R(x)))ByDeMorganslaw() - x 188 English Meaning: There was a patient who was not given both placebo ((P(x)) (Q(x) V (R(x))))ByDeMorganslaw(P V Q) - Pmedia 189 and medication. (P(x) (Q(x) V R(x)))ByDeMorganslaw(P) - P 190 P(x) (Q(x) V R(x))) 191 R.H.S L.H.S R.H.S A 192 \\item Every patient who took the placebo had migraines. (Hint: you will need to apply the conditional (P(x) V (Q(x) R(x))) (P(x) (Q(x) VR(x) identity, $p \\to q \\equiv \ eg p \\lor qs.)IIII 193 *Enter your answer below this comment line. 194 Logical Expression:-(x) (P(x) M(x)) 195 Negation:- (x) (P(x) M(x) ) (x) ( P(x) M(x)) (x) (P(x) M(x)> 196 English: There is some patient who was given placebo and was 197 not having migraines. 198 III1 199 \\item There is a patient who had migraines and was given the placebo. ! ! !\\ 200 *Enter your answer below this comment line. 201 Logical Expression:- (x) (M(x) P(x)) 202 Negation : - (x) (M(x) P(x) ) (x) (M(x) P(x)) (x) (M(x) P(x)) 203 English: Every patient who had migraines was not given the 204 placebo. 205 206 \\end { enumerate} 207 File outline 208 newpage