- WELCOME TO A LESSON ON TRUTH TABLES
FOR CONDITIONAL STATEMENTS.
HERE ARE THE MOST COMMON SYMBOLS USED FOR LOGIC AND TRUTH TABLES:
WE HAVE NEGATION OR NOT, IF-THEN, IF-AND-ONLY-IF,
CONJUNCTION OR AND, AND DISJUNCTION OR OR.
LET'S GO AND TAKE A LOOK AT OUR FIRST EXAMPLE.
HERE WE WANT TO COMPLETE THE TRUTH TABLE FOR P, Q,
AND IF P THEN Q.
SO THE FIRST THING WE HAVE TO DO
IS LIST ALL THE POSSIBLE COMBINATIONS OF TRUE AND FALSE
FOR P AND Q.
SINCE THERE ARE TWO OPTIONS FOR P, TRUE OR FALSE,
AND TWO OPTIONS FOR Q, TRUE AND FALSE,
THAT TELLS US WE NEED FOUR ROWS.
SO WE'LL START OFF BY LISTING P AS TRUE-TRUE FALSE-FALSE,
AND THEN Q COULD BE TRUE OR FALSE WHEN P IS TRUE,
AND Q COULD BE TRUE OR FALSE WHEN P IS FALSE.
SO WE HAVE FOUR POSSIBILITIES TO CONSIDER
WHEN LOOKING AT IF P THEN Q.
THERE'S A COUPLE WAYS OF MAKING SENSE OF THIS THIRD COLUMN.
ONE WAY IS TO THINK OF IF P THEN Q AS AN AGREEMENT
OR A PROMISE BY YOUR INSTRUCTOR
THAT IF YOU STUDY FOUR HOURS OR MORE
YOU WILL SCORE AN "A" ON THE EXAM.
AND THE ONLY TIME IF P THEN Q WOULD BE FALSE
IS WHEN THE INSTRUCTOR VIOLATES THIS AGREEMENT OR PROMISE.
SO FOR EXAMPLE, IF BOTH P AND Q ARE TRUE,
OF COURSE IF P THEN Q WOULD BE TRUE.
BUT IF P IS TRUE AND Q IS FALSE,
THAT MEANS YOU DO STUDY FOUR OR MORE HOURS
BUT YOU DON'T SCORE AN "A" ON THE EXAM,
AND THAT WOULD BE IN DIRECT VIOLATION
OF WHAT THE INSTRUCTOR PROMISED
AND THEREFORE IT WOULD BE FALSE.
AND THEN WHEN P IS FALSE AND Q IS TRUE,
THAT WOULD MEAN THAT YOU DON'T STUDY FOUR OR MORE HOURS
AND YOU STILL SCORE AN "A" ON THE EXAM.
WELL, AS SOON AS YOU KNOW
THAT YOU DON'T STUDY FOUR OR MORE HOURS,
THIS IS NOT GOING TO BE IN VIOLATION OF THE AGREEMENT
THAT THE INSTRUCTOR MADE,
AND THEREFORE IT'S TRUE.
AND THE SAME THING FOR WHEN P IS FALSE AND Q IS FALSE--
THAT WOULD MEAN YOU DON'T STUDY FOUR OR MORE HOURS
AND YOU DON'T SCORE AN "A."
AGAIN, THE INSTRUCTOR ONLY MADE AN AGREEMENT
IF YOU STUDY FOUR OR MORE HOURS.
SO IN THIS CASE IT'S NOT IN VIOLATION OF THE AGREEMENT
AND THEREFORE IS TRUE.
NOTICE HOW THERE'S ONLY ONE FALSE IN THIS COLUMN,
AND THAT'S IMPORTANT TO RECOGNIZE.
THE CONDITIONAL STATEMENT IF P THEN Q IS ONLY FALSE
WHEN THE HYPOTHESIS IS TRUE AND THE CONCLUSION IS FALSE.
EVERYTHING ELSE IS TRUE.
AND IF WE CAN REMEMBER THAT,
IT MAKES THESE TABLES A LOT EASIER TO COMPLETE.
LET'S TAKE A LOOK AT THE NEXT EXAMPLE.
WE WANT TO COMPLETE A TRUTH TABLE FOR NOT P, NOT Q,
AND IF NOT P THEN NOT Q.
AND THIS IS ACTUALLY THE INVERSE OF IF P THEN Q.
SO WE'LL START BY LISTING P AND Q
AS WE DID ON THE PREVIOUS SLIDE,
AND THEN NOTICE THAT P AND NOT P ARE JUST OPPOSITES,
MEANING WHEN P IS TRUE NOT P IS FALSE,
AND WHEN P IS FALSE NOT P IS TRUE.
AND THE SAME THING FOR Q AND NOT Q.
SO TO COMPLETE THIS LAST COLUMN,
NOW WE'RE ONLY GOING TO LOOK AT THE THIRD AND FOURTH COLUMNS.
AND THE QUICKEST WAY TO DO THIS IS,
AGAIN, TO RECOGNIZE THAT THIS CONDITIONAL STATEMENT
IS ONLY GOING TO BE FALSE
WHEN THE HYPOTHESIS, NOT P, IS TRUE
AND THE CONCLUSION, NOT Q, IS FALSE.
AND THAT OCCURS RIGHT HERE.
NOTICE THAT EVERYTHING ELSE IS TRUE.
NOW OF COURSE WE COULD MAKE THE ANALOGY LIKE WE DID
ON THE PREVIOUS SLIDE
BY SETTING IT UP AS AN AGREEMENT WITH YOUR INSTRUCTOR,
HOWEVER THIS CAN GET A LITTLE BIT CONFUSING
BECAUSE IF YOU SAY THAT THE NEGATION OF P IS FALSE,
THAT ACTUALLY MEANS P WOULD BE TRUE AND SO ON.
SO IT IS A LITTLE BIT MORE DIFFICULT TO MAKE SENSE OF,
BUT IT CAN BE DONE.
LET'S GO AHEAD AND TAKE A LOOK AT ANOTHER EXAMPLE.
HERE WE'LL TAKE A LOOK AT THE CONVERSE OF IF P THEN Q.
SO WE'RE GOING TO LOOK AT P, Q, AND IF Q THEN P.
SO WE'LL LIST P AND Q AGAIN AS WE DID.
AND NOW WE JUST HAVE TO BE CAREFUL,
AND WE'RE TAKING A LOOK AT IT IN THE OPPOSITE ORDER,
MEANING IF WE WANT TO ANALYZE IF Q THEN P,
WE'RE TAKING A LOOK AT IT IN THIS ORDER HERE.
SO YOU MAY ACTUALLY FIND IT HELPFUL
TO SWITCH THE ORDER OF THE COLUMNS
TO MAKE IT MATCH THE IF-THEN STATEMENT
IF YOU FIND THAT EASIER.
AND AGAIN, IF I USE THE FACT THAT IF Q THEN P
IS ONLY GOING TO BE FALSE WHEN Q IS TRUE AND P IS FALSE,
WE CAN COMPLETE THIS COLUMN RATHER QUICKLY.
AGAIN, READING IT FROM RIGHT TO LEFT,
NOTICE THAT Q IS TRUE AND P IS FALSE IN THIS COLUMN HERE,
WHICH LEADS TO THE FALSE IN THIS THIRD COLUMN.
EVERYTHING ELSE, ONCE AGAIN, IS JUST TRUE.
NOW LET'S GO AHEAD AND TRY ONE MORE.
HERE WE'LL TAKE A LOOK AT THE CONTRAPOSITIVE OF IF P THEN Q,
WHICH MEANS WE'LL TAKE A LOOK AT IF NOT Q THEN NOT P.
SO AGAIN WE'LL LIST P, Q, NOT P, NOT Q,
AND THEN THE LAST COLUMN IS ALWAYS OUR CONDITIONAL.
SO NOTICE HOW IN THIS ONE
I DID SWITCH THE ORDER OF NOT Q AND NOT P
JUST TO MAKE THE ANALYSIS A LITTLE BIT EASIER.
AND AGAIN WHAT I'M LOOKING FOR IS WHEN THE HYPOTHESIS IS TRUE
AND THE CONCLUSION IS FALSE.
SO I'M LOOKING FOR TRUE-FALSE AND HERE IT IS,
AND AGAIN, EVERYTHING ELSE IN THIS COLUMN IS TRUE.
I HOPE YOU FOUND THESE EXAMPLES HELPFUL.
THANK YOU FOR WATCHING.