Logic Conditional proof How do I solve these problems 1 Q P

Logic Conditional proof? How do I solve these problems?

1) -Q -> (-P -> Q), -Q -P -> Q

2) P -> (Q -> R), P -> Q, P R

Solution

1………………. LET ….. R = (- P IMPLIES Q)……………………………..1 S = -Q IMPLIES R …………………………….2 SO IT IS GIVEN THAT -Q IMPLIES R AS PER 2 …. WE ARE TO PROVE THAT -Q TURNSTILE R…………………………………3 PLEASE NOTE THAT IN GENERAL ….X TURNSTILE Y MEANS , THAT WHENVER X IS TRUE , THEN Y IS ALSO TRUE …… SO 3 IS NOTHING BUT TO PROVE THAT ... FROM -Q WE KNOW THAT R IS TRUE …. THIS IS ALREADY PROVED BY THE IMPLICATION STATEMENT GIVEN UNDER 2 SINCE IMPLICATION STATEMENT X IMPLIES Y MEANS THAT IF X IS TRUE THEN Y SHALL BE TRUE . HENCE WE SAY -Q TURNSTILE R …..PROVED …. 2…… LET …… S= ( Q IMPLIES R )…………………………………….4 T = P IMPLIES S ……………………………….5 U = P IMPLIES Q ……………………………..6 SO IT IS GIVEN THAT P IMPLIES S AS PER 5 AND P IMLIES Q AS PER 6 WE ARE TO PROVE THAT U,P TURNSTILE R …………………………………6 PLEASE NOTE THAT IN GENERAL ….X TURNSTILE Y MEANS , THAT WHENVER X IS TRUE , THEN Y IS ALSO TRUE …… SO 6 IS NOTHING BUT TO PROVE THAT ... FROM U AND P WE KNOW THAT R IS TRUE …. THAT IS TO PROVE THAT IF U & P ARE TRUE THEN R IS TRUE THAT IS TO PROVE THAT IF U * P [ INTERSECTION OF U & P ] IS TRUE THEN R IS TRUE FROM THE TRUTH TABLE GIVEN BELOW , WE FIND THAT IN S.NO.2 , WHEN P IS TRUE , Q IS TRUE WE GET U*P AS TRUE ..BUT R IS NOT TRUE .. SO THE GIVEN PROPOSITION IS NOT CORRECT .. PLEASE CHECK FOR TYPO IF ANY & COME BACK HOPE YOU UNDERSTOOD THE METHODOLOGY. IF IN DOUBT PLEASE COME BACK 1…MEANS …TRUE …..; …..0…... MEANS…….FALSE TRUTH TABLE S.NO. P Q R S T U U * P CONCLUSION 1 1 1 1 1 1 1 1 OK 2 1 1 0 0 0 1 1 R IS NOT TRUE ..NOT MATCHING 3 1 0 1 1 1 0 0 4 1 0 0 1 1 0 0 5 0 1 1 1 1 1 0 6 0 1 0 0 1 1 0 7 0 0 1 1 1 1 0 8 0 0 0 1 1 1 0