- WELCOME TO A LESSON ON TRUTH TABLES
FOR THE BICONDITIONAL STATEMENT.
AGAIN, HERE ARE THE SYMBOLS OFTEN USED
FOR LOGIC AND TRUTH TABLES.
YOU MAY WANT TO PAUSE THE VIDEO AND TAKE A LOOK AT THESE;
WE'VE SEEN THESE BEFORE.
BUT LET'S TAKE A MOMENT AND REVIEW
IF-AND-ONLY-IF STATEMENTS, ALSO CALLED BICONDITIONAL STATEMENTS.
IF A STATEMENT IS TRUE, MEANING IF P THEN Q IS TRUE,
AND THE CONVERSE IS TRUE, MEANING IF Q THEN P IS TRUE,
WE CAN STATE P IF AND ONLY IF Q IN ANY OF THESE THREE WAYS.
AN EXAMPLE OF THIS WOULD BE,
"IF TWO LINES ARE PERPENDICULAR THEN THEY FORM A RIGHT ANGLE,"
WHICH IS TRUE.
AND IF THIS IS IN THE FORM IF P THEN Q, IF Q THEN P WOULD BE,
"IF TWO LINES FORM A RIGHT ANGLE
THEN THE LINES ARE PERPENDICULAR,"
WHICH IS ALSO TRUE.
SO THIS WOULD BE AN EXAMPLE OF AN IF-AND-ONLY-IF STATEMENT.
SO WE COULD STATE THAT "TWO LINES ARE PERPENDICULAR
IF AND ONLY IF THEY FORM A RIGHT ANGLE."
BUT WHEN COMPLETING A TRUTH TABLE,
THE MOST IMPORTANT THING TO REMEMBER
IS THIS BICONDITIONAL STATEMENT IS TRUE IF P THEN Q IS TRUE,
AND IF Q THEN P IS TRUE.
LET'S TAKE A LOOK AT OUR EXAMPLE.
HERE WE WANT TO COMPLETE THE TRUTH TABLE FOR P, Q, AND P
IF AND ONLY IF Q.
SO TO DO THIS WHAT WE'RE GOING TO DO IS CREATE A TABLE
WITH P, Q, IF P THEN Q, AND THEN IF Q THEN P.
AND REMEMBER THAT THE BICONDITIONAL STATEMENT
IS GOING TO BE TRUE ONLY WHEN
BOTH THESE CONDITIONAL STATEMENTS ARE TRUE.
AND SO BY NOW WE SHOULD BE PRETTY FAMILIAR THAT FOR P AND Q
WE'RE GOING TO HAVE TRUE-TRUE,
TRUE-FALSE, FALSE-TRUE, AND FALSE-FALSE,
FOR THE POSSIBILITIES OF P AND Q.
AND PROBABLY THE MOST IMPORTANT THING TO REMEMBER HERE IS THAT
A CONDITIONAL STATEMENT, IF P THEN Q,
IS ONLY FALSE WHEN THE HYPOTHESIS IS TRUE
AND THE CONCLUSION IS FALSE.
SO TAKING A LOOK AT THIS STATEMENT HERE,
IF P THEN Q, WE'LL IDENTIFY WHEN THE HYPOTHESIS IS TRUE
AND THE CONCLUSION IS FALSE,
BECAUSE THAT'S THE ONLY TIME
WHEN THIS CONDITIONAL WILL BE FALSE.
AND NOTICE THAT OCCURS RIGHT HERE--WE HAVE TRUE-FALSE,
AND THEREFORE THE CONDITIONAL IS FALSE.
EVERYTHING ELSE IS TRUE.
AND NOW WE'RE GOING TO DO THE SAME FOR IF Q THEN P,
LOOKING AT THE FIRST TWO COLUMNS IN THE REVERSE ORDER.
TO IDENTIFY WHERE IF Q THEN P IS GOING TO BE FALSE,
WE WANT TO IDENTIFY WHERE Q IS NOW TRUE AND P IS FALSE.
SO LOOKING AT THESE TWO COLUMNS IN THE OPPOSITE DIRECTION,
WE CAN SEE THAT Q IS TRUE AND P IS FALSE HERE,
AND THEREFORE IN THIS ROW IF Q THEN P IS FALSE
AND EVERYTHING ELSE IS TRUE.
IT'S NOT A COMPLETE -- COLUMN
FOR THE BICONDITIONAL STATEMENT P IF AND ONLY IF Q;
WE NEED TO FIND WHERE BOTH THE THIRD AND FOURTH COLUMN
ARE TRUE.
AND NOTICE THAT OCCURS HERE AND IT ALSO OCCURS HERE.
THE OTHER TWO ROWS WE HAVE FALSE-TRUE AND TRUE-FALSE;
AND AGAIN, THEY BOTH HAVE TO BE TRUE
IN ORDER FOR THE BICONDITIONAL STATEMENT TO BE TRUE.
SO I HOPE THAT HELPS EXPLAIN THE TRUTH TABLE
FOR A BICONDITIONAL STATEMENT.
THANK YOU FOR WATCHING.