- WELCOME TO A LESSON ON SHOWING STATEMENTS ARE EQUIVALENT.
WE'VE SEEN MOST OF THESE SYMBOLS BEFORE IN PREVIOUS VIDEOS--
THEY'RE USED FOR LOGIC AND TRUTH TABLES.
THE ONLY NEW SYMBOL HERE IS THE LAST ONE,
WHERE IF TWO STATEMENTS ARE EQUIVALENT
WE CAN USE THIS SYMBOL HERE.
LET'S GO AND TAKE A LOOK AT OUR FIRST EXAMPLE.
WE WANT TO SHOW THAT P OR NOT Q
AND IF NOT P THEN NOT Q ARE EQUIVALENT.
SO TO SET THIS UP, THE FIRST COLUMNS WILL BE P, Q, NOT P,
AND NOT Q.
AND THESE SHOULD BE PRETTY STRAIGHTFORWARD TO SET UP,
WE'VE DONE THIS SEVERAL TIMES NOW.
NOTICE THAT WHERE P IS TRUE, NOT P IS FALSE,
AND WHERE P IS FALSE, NOT P IS TRUE.
AND THE SAME THING FOR Q AND NOT Q.
SO NOW WE'RE GOING TO TAKE A LOOK AT THIS FIFTH COLUMN--
P OR NOT Q.
REMEMBER, FOR THIS TO BE TRUE,
ONLY ONE OF THESE HAS TO BE TRUE,
OR WE CAN SAY IT'LL ONLY BE FALSE WHEN THEY'RE BOTH FALSE.
SO WE'RE GOING TO ANALYZE THE FIRST COLUMN, P,
AND THE FOURTH COLUMN, NOT Q.
SO WE HAVE TRUE-FALSE, WHICH IS TRUE
BECAUSE ONE OF THEM IS TRUE, WE HAVE TRUE-TRUE, WHICH IS TRUE,
FALSE-FALSE IS THE ONLY TIME THIS STATEMENT WILL BE FALSE,
AND FALSE-TRUE IS TRUE.
AND NOW WE'RE GOING TO GO TO THE LAST COLUMN AND TRY TO DETERMINE
WHERE IF NOT P THEN NOT Q IS TRUE AND FALSE.
REMEMBER, FOR A CONDITIONAL STATEMENT LIKE THIS,
IT'S ONLY GOING TO BE FALSE WHEN THE HYPOTHESIS IS TRUE
AND THE CONCLUSION IS FALSE.
SO WE'RE GOING TO TAKE A LOOK AT THE COLUMN NOT P
AND LOOK TO SEE WHERE IT'S TRUE
AND THEN DETERMINE WHERE, AT THE SAME TIME, NOT Q IS FALSE.
THAT'S THE ONLY TIME THIS CONDITIONAL STATEMENT
WILL BE FALSE.
SO HERE IS NOT P AND HERE IS NOT Q.
AND AGAIN, WE'RE LOOKING FOR TRUE-FALSE, AND HERE IT IS.
NOTICE THAT WHEN NOT P IS TRUE AND NOT Q IS FALSE
THIS CONDITIONAL IS FALSE, AND EVERYWHERE ELSE IT'S TRUE.
AND NOW WE'VE SHOWN THESE TWO STATEMENTS ARE EQUIVALENT
BECAUSE NOTICE HOW THEY MATCH UP PERFECTLY--
TRUE, TRUE, FALSE, TRUE.
SO WE CAN SAY THAT P OR NOT Q IS EQUIVALENT TO
IF NOT P THEN NOT Q.
LET'S TAKE A LOOK AT ONE MORE.
HERE WE WANT TO SHOW THAT P OR Q
AND IF NOT Q THEN P ARE EQUIVALENT.
SO NOTICE THIS ONE DOES NOT REQUIRE NOT P,
SO WE HAVE P, Q, AND NOT Q.
SO FOR P OR Q WE'LL TAKE A LOOK AT THE FIRST TWO COLUMNS.
AND AGAIN, IT'S AN OR,
SO IT ONLY REQUIRES ONE OF THESE TO BE TRUE,
OR WE COULD SAY THAT IT WILL ONLY BE FALSE
WHEN THEY'RE BOTH FALSE.
HERE IS WHERE THEY'RE BOTH FALSE,
AND NOTICE THIS IS THE ONLY TIME P OR Q IS FALSE,
EVERYWHERE ELSE IT'S TRUE.
AND NOW WE'RE GOING TO TAKE A LOOK AT IF NOT Q THEN P.
WE NEED TO BE A LITTLE CAREFUL ABOUT THIS ONE
BECAUSE WE'RE GOING TO BE LOOKING AT THESE COLUMNS
IN THE OPPOSITE DIRECTION.
HERE'S OUR HYPOTHESIS, NOT Q, AND HERE'S P.
SO LOOK AT THIS FROM RIGHT TO LEFT.
AND AGAIN, THE ONLY TIME THIS IS GOING TO BE FALSE
IS WHEN NOT Q IS TRUE AND P IS FALSE.
NOT Q IS TRUE HERE AND P IS FALSE HERE,
WHICH IS THE ONLY TIME THIS CONDITIONAL IS FALSE,
EVERYTHING ELSE IS TRUE.
AND NOW IF WE TAKE A LOOK AT THESE LAST TWO COLUMNS,
NOTICE WE HAVE TRUE-TRUE, TRUE-TRUE, TRUE-TRUE,
AND FALSE-FALSE,
THEREFORE WE'VE SHOWN THESE STATEMENTS ARE EQUIVALENT.
P OR Q IS EQUIVALENT TO IF NOT Q THEN P.
I HOPE YOU FOUND THIS HELPFUL.