TUGAS 3 PDM

BUKTIKAN KESAHAN ATURAN PENYIMPULAN BERIKUT:
1. Silogisme
p→q
q→r
.˙. p→r

(p→q)Λ(q→r) →(p→r)
≡[~(p→q)v~(q→r)]v(p→r) (imp)
≡[~(~pvq)v~(~qvr)]v(~pvr) (imp)
≡[(pΛ~q)v(qΛ~r)]v(~pvr) (DM)
≡{[( pΛ~q)vq]Λ[(pΛ~q)v~r]} v(~pvr) (dist)
≡{[( pvq)Λ(~qvq)]Λ[(pv~r)Λ(~qv~r]} v(~pvr) (dist)
≡{[( pvq)ΛT]Λ[(pv~r)Λ(~qv~r]} v(~pvr) (komp)
≡[( pvq)Λ(pv~r)Λ(~qv~r)]v(~pvr) (id)
≡{[(pvq)v~p]Λ[(pv~r)v~p]Λ[(~qv~r)v~p]}vr (dist)
≡{[(pv~p)vq]Λ[(pv~p)v~r)]Λ[(~qv~r)]}v r (asso)
≡[( T vq)Λ(Tv~r)Λ[(~qv~r)]}v r (komp)
≡ [(TΛTΛ(~qv~r)]vr (id)
≡(~qv~r)vr (id)
≡~qv(~rvr) (asso)
≡~qv T (komp)
≡ T (id)

2. Distruktive Silogisma (DS)
pvq
~p
.˙. q
[(pvq)Λ~p]→q
≡(pΛ~p)v(qΛ~p)→q (dist)
≡ Fv(qΛ~p)→q (komp)
≡ (qΛ~p)→q (id)
≡ ~(qΛ~p)vq (imp)
≡ (~qvp)vq (DM)
≡ (pv~q)vq (kom)
≡ pv(~qvq) (asso)
≡ pv(qv~q) (kom)
≡ pvT (komp)
≡ T (id)

3. Konstruktive Dilema (KD)
(p→q)Λ(r→s)
(pvr)
.˙.qvs
{[(p→q)Λ(r→s)]Λ(pvr)}→( qvs)
≡[(~pvq)Λ(~rvs)Λ(pvr)]→(qvs) (imp)
≡[(pΛ~q)v(rΛ~s)v(~pΛ~r)]v(qvs) (imp)
≡[( pΛ~q)v(rΛ~s)v(qvs)]v(~pΛ~r) (asso)
≡{[( pΛ~q)vq]v[(rΛ~s)vs]}v(~pΛ~r) (asso)
≡{[qv( pΛ~q)]v[sv(rΛ~s)]} v(~pΛ~r) (kom)
≡{[(qvp)Λ(qv~q]v[(svr)Λ(sv~s)]}v(~pΛ~r) (dist)
≡[(qvp)ΛT]v[(svr)ΛT]v(~pΛ~r) (komp)
≡[(qvp)v(svr)]v(~pΛ~r) (id)
≡(qvpvsvr)v(~pΛ~r) (asso)
≡(qvs)v(pvr) v(~pΛ~r) (asso)
≡(qvs)v[(pvr)v~(pvr)] (DM)
≡(qvs)vT (komp)
≡ T (id)

4. Distruktif Delema (KD)
(p→q)Λ(r→s)
~qv~s
.˙.~pv~r
{[(p→q)Λ(r→s)]Λ(~qv~s)}→ (~pv~r)
≡[(~pvq)Λ(~rvs)Λ(~qv~s)}→ (~pv~r) (imp)
≡[(pΛ~q)v(rΛ~s)v(qΛs)]v(~pv~r) (imp)
≡[(pΛ~q)v(qΛs)]v[(rΛ~s)v(~pv~r)] (asso)
≡{[(pΛ~q)vq]Λ[(pΛ~q)vs]v[(rΛ~s)v(~pv~r)]} (asso)
≡{[(pvq)Λ(~qvq)]Λ[(pvs)Λ(~qvs)]v[(rΛ~s)v(~pv~r)]} (dist)
≡{[(pvq)Λ(~qvq)Λ(pvs)Λ(~qvs)]}v{[(rΛ~s)v~r]v~p} (asso)
≡{[(pvq)Λ(~qvq)Λ(pvs)Λ(~qvs)]}v{[(rv~r)Λ(~sv~r)]v~p} (dist)
≡{[(pvq)Λ T Λ(pvs)Λ(~qvs)]}v{[ T Λ(~sv~r)]v~p} (komp)
≡[(pvq)Λ(pvs)Λ(~qvs)]v[(~sv~r)v~p] (id)
≡[(pvq)Λ(pvs)Λ(~qvs)v~p]v(~sv~r) (asso)
≡[(pvq)Λ(pvs)Λ(~qvs)v~p]v(~sv~r) (asso)
≡[(pvq v~p)Λ(pvs v~p)Λ(~qvs v~p)]v(~sv~r) (dist)
≡{[(pv~p)vq]Λ[(pv~p)vs]Λ(~qvs v~p)]v(~sv~r) (asso)
≡[( T vq)Λ( T vs)Λ(~qvs v~p)]v(~sv~r) (komp)
≡(TΛ T)Λ(~qvs v~p)v(~sv~r) (id)
≡(~qvs v~p)v(~sv~r) (id)
≡(sv~sv)v(~pv~qv~r) (asso)
≡ T v (~pv~qv~r) (komp)
≡ T (id)