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 209 210 PROBLEM 2 snhu_logo.png 211 Use De Morgan's law for quantified statements and the laws of propositional 212 - \\section*{Problem 2} logic to show the following equivalences: 213 214 Use De Morgan's law for quantified statements and the laws of propositional logic to show the following (a) WE (P(2) A-Q(1)) = Hr (-P(1) VQ(2)) A equivalences :\\\\ 215 + \\begin{enumerate} [label=(\\alph*) ] (b) 7VI (-P(I) + Q(1)) = 1: (-P(I) A-Q(I)) 216 \\item $\ eg \\forall x \\, \\left(P(x) \\land \ eg Q(x) \ ight) \\ \\equiv \\ \\exists x \\, \\left(\ eg P(x) \\lor Q(x) \ ight) $\\Ill Given ( P(x) Q(x)) (P(x) Q(x)) 217 *Enter your answer below this comment line. Take L.H.S (P(x) Q(x)) 218 (P(x))V Q(x))LawofImplies(P) - P V Q (P(x) V Q(x))ByDeMorgan( 219 slaw(P) - P X 220 IIII (P(x)V Q(x))ByDeMorgan0 221 \\item $\ eg \\forall x \\, \\left(\ eg P(x) \\to Q(x) \ ight) \\; \\equiv \\ \\exists x \\, \\left(\ eg P(x) \\land slaw() - X X \ eg Q(x) \ ight) SIIll (P(x) Q(x))ByDeMorgan0 222 *Enter your answer below this comment line. slaw(P V Q) - Pmedia (P(x) Q(x)) x 223 Given ( P(x) - Q(x)) = (P(x) A Q(x)) R.H.S 224 L.H.S R.H.S x 225 Take L. H. S = (P(x) - Q(x)) 226 X 227 = ((P(x) )V Q(x)) LawofImplies (P) = P V Q (c) -3x ( P(I) V (Q(I) A -R(I)) ) = VI(P(I) A(-Q(I) V R(I))) 228 x 229 = (P(x) V Q(x) ) ByDeMorgan0 (P(x) V (Q(x) R(x))) (P(x) (Q(x) V R(x))) 230 Take L.H.S (P(x) V (Q(x) R(x))) 231 slaw(P) = P (P(x) V (Q(x) R(x)))ByDeMorganslaw() - x 232 ((P(x)) (Q(x) V (R(x))))ByDeMorganslaw(P V Q) - Pmedia (P(x) (Q(x) V R(x)))ByDeMorganslaw(P) - P 233 = (P(x)V Q(x) ) ByDeMorgan0 P(x) (Q(x) V R(x))) 234 R.H.S x 235 slaw() = 3x L.H.S R.H.S 236 (P(x) V (Q(x) R(x))) (P(x) (Q(x) VR(x) x 237 = (P(x) A Q(x) ) ByDeMorgan0 238 239 slaw(P V Q) = Pmedia 240 x 241 = (P(x) A Q(x)) 242 X 243 = R. H. S 244 x 245 L. H. S = R. H. S 246 III1 247 \\item $\ eg \\exists x \\, \\big(\ eg P(x) \\lor \\left(Q(x) \\land \ eg R(x) \ ight) \\big) \\ \\equiv \\ \\forall x 1, \\big( P(x) \\land \\left( \ eg Q(x) \\lor R(x) \ ight) \\big)$!!!\\ 248 %Enter your answer below this comment line. 249 x 250 (P(x) V (Q(x) A R(x))) = (P(x) A (Q(x) V R(x)> > File outline 251