- WELCOME TO A LESSON ON CONVERSE, CONTRAPOSITIVE,
AND INVERSE OF AN IF-THEN STATEMENT.
IT CAN BE HELPFUL TO MODEL A CONDITIONAL STATEMENT
BY USING A VENN DIAGRAM.
WE CAN DRAW A CIRCLE TO REPRESENT THE STATEMENT P,
AS WE SEE HERE.
SO IF WE'RE INSIDE P, P IS TRUE.
AND IF WE'RE OUTSIDE P, P WOULD BE FALSE,
OR WE COULD SAY "NOT P."
SO FOR THE CONDITIONAL IF P THEN Q,
THE VENN DIAGRAM WOULD HAVE SET P INSIDE SET Q,
GUARANTEEING THAT IF P IS TRUE, Q WOULD ALSO BE TRUE.
FOR THE CONVERSE,
IF WE SWITCH THE ORDER OF THE HYPOTHESIS AND CONCLUSION,
WE WOULD HAVE IF Q THEN P.
NOTICE HOW WE ALSO SWITCHED THE SETS P AND Q,
SO IF Q IS TRUE IT'S GUARANTEED THAT P WOULD ALSO BE TRUE.
AND FOR THE BICONDITIONAL, P IF AND ONLY IF Q.
REMEMBER, THAT MEANS IF P THEN Q IS TRUE,
AND IF Q THEN P IS TRUE,
P AND Q WOULD ACTUALLY BE THE SAME SET.
SO IF WE'RE IN P WE'RE ALSO IN Q.
NOW LET'S TALK ABOUT THE CONVERSE, INVERSE,
AND CONTRAPOSITIVE OF A GIVEN CONDITIONAL STATEMENT.
IF WE WANT TO MODEL THE CONDITIONAL STATEMENT,
"IF I WAKE UP EARLY THEN I WILL LIFT WEIGHTS,"
ASSUMING THIS IS IN THE FORM OF IF P THEN Q,
WE'D HAVE TO HAVE THE SET P INSIDE SET Q.
SO IF WE'RE INSIDE P, MEANING WE WAKE UP EARLY,
THEN WE'LL ALSO BE INSIDE SET Q, MEANING WE'LL LIFT WEIGHTS.
SO FOR THE CONVERSE IF Q THEN P,
WE INTERCHANGE THE HYPOTHESIS AND CONCLUSION.
SO WE'D HAVE, "IF I LIFT WEIGHTS THEN I WAKE UP EARLY."
LOOKING AT THE VENN DIAGRAM, "IF I LIFT WEIGHTS"
MEANS I'M INSIDE SET Q,
"THEN I WAKE UP EARLY" MEANS I'M INSIDE SET P.
SO JUST BECAUSE I'M INSIDE Q DOESN'T GUARANTEE
THAT I'M INSIDE P,
THEREFORE THIS IS NOT LOGICALLY EQUIVALENT TO IF P THEN Q,
WHICH MEANS IT MAY BE TRUE OR IT MAY BE FALSE.
THE CONTRAPOSITIVE WOULD BE IF NOT Q THEN NOT P.
SO WE INTERCHANGE THE HYPOTHESIS AND CONCLUSION
AND ALSO NEGATE THEM.
SO THE STATEMENT WOULD BE,
"IF I DON'T LIFT WEIGHTS THEN I DON'T WAKE UP EARLY."
AGAIN, LOOKING AT THE VENN DIAGRAM,
"IF I DON'T LIFT WEIGHTS"--
AND THIS MEANS I'M NOT INSIDE Q OR THAT I AM OUTSIDE Q--
SO IF I'M OUTSIDE Q THEN "I DON'T WAKE UP EARLY,"
WHICH MEANS I'M OUTSIDE P.
SO IF I'M OUTSIDE Q I'M GUARANTEED TO BE OUTSIDE P,
THEREFORE THIS STATEMENT IS LOGICALLY EQUIVALENT
TO IF P THEN Q.
AND THEN FOR THE INVERSE WE WOULD HAVE IF NOT P THEN NOT Q.
SO WE NEGATE THE HYPOTHESIS AND CONCLUSION,
SO WE'D HAVE "IF I DON'T WAKE UP EARLY I WILL NOT LIFT WEIGHTS."
SO "IF I DON'T WAKE UP EARLY" MEANS I'M OUTSIDE P.
SO IF I'M OUTSIDE P THEN "I WILL NOT LIFT WEIGHTS"
OR I'M OUTSIDE Q.
SO IF I'M OUTSIDE P I'M NOT GUARANTEED TO BE OUTSIDE Q,
AND THEREFORE THIS IS NOT LOGICALLY EQUIVALENT TO
IF P THEN Q,
WHICH MEANS IT MAY BE TRUE OR FALSE.
NOW LET'S TAKE A LOOK AT A SECOND EXAMPLE.
HERE WE WANT TO USE THE GIVEN STATEMENT
TO DETERMINE THE CONVERSE, CONTRAPOSITIVE, AND INVERSE,
AND THEN DETERMINE IF THE NEW STATEMENT IS EQUIVALENT,
AND WE'LL ALSO DETERMINE IF IT WOULD BE TRUE OR FALSE.
THE GIVEN STATEMENT IS,
"IF I'M HIKING IN THE GRAND CANYON THEN I'M IN ARIZONA."
THE GRAND CANYON IS ONE OF THE LARGEST TOURIST ATTRACTIONS
LOCATED IN ARIZONA,
AND IT'S ALSO A NATIONAL LANDMARK.
SO IF I'M HIKING THE GRAND CANYON COULD BE MODELED
BY THIS CIRCLE P,
AND IF I'M IN ARIZONA WOULD BE MODELED BY CIRCLE Q.
NOTICE HOW IT'S POSSIBLE TO BE IN ARIZONA
BUT NOT BE HIKING THE GRAND CANYON,
MEANING OUTSIDE P BUT INSIDE Q.
SO THE CONVERSE IF Q THEN P WOULD BE,
"IF I'M IN ARIZONA THEN I'M HIKING THE GRAND CANYON."
SO IF I'M INSIDE SET Q I'M NOT GUARANTEED TO BE INSIDE P,
AND THEREFORE THE CONVERSE IS NOT EQUIVALENT TO IF P THEN Q.
SO IN THIS SPECIFIC CASE, JUST BECAUSE I'M IN ARIZONA
DOES NOT MEAN THAT I'M HIKING THE GRAND CANYON,
THEREFORE THIS WOULD BE FALSE.
THE CONTRAPOSITIVE IS IF NOT Q THEN NOT P,
AND THIS WOULD BE, "IF I'M NOT IN ARIZONA
THEN I'M NOT HIKING THE GRAND CANYON."
"IF I'M NOT IN ARIZONA" WOULD MEAN THAT I'M OUTSIDE Q,
AND IF I'M OUTSIDE Q I'M GUARANTEED TO BE OUTSIDE P,
THEREFORE THE CONTRAPOSITIVE IS LOGICALLY EQUIVALENT TO
IF P THEN Q,
THEREFORE THIS IS GUARANTEED TO BE TRUE.
THE INVERSE WOULD BE IF NOT P THEN NOT Q.
SO "IF I'M NOT HIKING THE GRAND CANYON
THEN I'M NOT IN ARIZONA."
IF I'M NOT HIKING THE GRAND CANYON,
THAT WOULD MEAN I'M OUTSIDE P.
SO IF I'M OUTSIDE P
THEN "I'M NOT IN ARIZONA" WOULD BE OUTSIDE Q.
WELL, JUST BECAUSE I'M OUTSIDE P DOESN'T MEAN I'M OUTSIDE Q.
SO THIS IS NOT LOGICALLY EQUIVALENT TO IF P THEN Q.
FOR THIS SPECIFIC CASE,
"IF I'M NOT HIKING THE GRAND CANYON THEN I'M NOT IN ARIZONA"
WOULD BE FALSE
BECAUSE THERE ARE A LOT OF OTHER PLACES TO BE IN ARIZONA
OTHER THAN THE GRAND CANYON.
FOR EXAMPLE, IF I'M IN PHOENIX I'M NOT HIKING THE GRAND CANYON
BUT I AM IN ARIZONA.
AND JUST IN CASE YOU HAVEN'T SEEN THE GRAND CANYON BEFORE,
HERE'S A NICE PICTURE OF IT.
AND TO HIKE TO THE BOTTOM AND BACK UP
TAKES MOST PEOPLE THE ENTIRE DAY.
I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING.