- WELCOME TO A LESSON ON SUBSETS.
THE GOALS OF THIS VIDEO ARE TO DEFINE A SUBSET
AND PROPER SUBSET, AND THEN ALSO USE
THE PROPER NOTATION FOR BOTH.
IF EVERY MEMBER OF SET "A" IS ALSO A MEMBER OF SET B,
THEN "A" IS A SUBSET OF B, AND WE USE THIS NOTATION HERE.
WE CAN ALSO SAY THAT "A" IS CONTAINED IN B.
AND THERE IS AN ALTERNATE NOTATION.
IF WE SWITCH THE ORDER OF THE NOTATION,
WE CAN SAY THAT B IS A SUPERSET OF "A,"
OR THAT B INCLUDES "A."
NOTICE USING EITHER NOTATION,
THE SUBSET SYMBOL OPENS TOWARDS THE B OR POINTS
TOWARD THE "A" IN EITHER CASE.
IF "A" IS NOT A SUBSET OF B WE CAN USE THIS NOTATION HERE,
WHERE I PUT A SLASH THROUGH THE SUBSET SYMBOL.
SO LOOKING AT SETS "A," B, AND C,
NOTICE THAT ALL OF THE MEMBERS OF SET C ARE ALSO MEMBERS
OF SET "A."
ONE IS IN BOTH SET C AND "A," TWO IS IN SET C AND "A,"
AND EIGHT IS IN SET C AND "A."
SO WE CAN SAY THAT C IS A SUBSET OF A,
OR WE CAN SAY "A" IS A SUPERSET OF SET C.
NOW LOOKING AT SET B AND C FOR A MOMENT,
NOTICE THAT SET B CONTAINS TWO AND EIGHT,
BUT DOES NOT CONTAIN THE NUMBER ONE.
THEREFORE WE CAN SAY THAT C IS NOT A SUBSET OF SET B.
IF "A" IS A SUBSET OF B, BUT "A" IS NOT = B,
WE CAN SAY THAT "A" IS A PROPER SUBSET OF B,
WRITTEN USING THIS NOTATION HERE.
SO NOTICE THAT IT'S THE SAME SYMBOL
AS THE SUBSET SYMBOL EXCEPT THAT THE BAR
UNDERNEATH THE SYMBOL IS OMITTED,
OR WE CAN INCLUDE THE BAR AND PUT A SLASH THROUGH IT
SHOWING THAT THE TWO SETS ARE NOT = EACH OTHER.
TAKING A LOOK AT THESE THREE SETS HERE,
LET'S TAKE A LOOK AT SET "A" AND SET B FIRST.
NOTICE THAT ALL OF THE ELEMENTS IN B ARE ALSO IN SET "A,"
BUT THEY'RE NOT EQUAL SETS.
SO WE CAN SAY THAT B IS A SUBSET OF "A."
AND SINCE THE TWO SETS ARE NOT EQUAL TO EACH OTHER,
WE CAN ALSO SAY THAT B IS A PROPER SET OF "A."
BUT NOW IF WE TAKE A LOOK AT SETS "A" AND C,
NOTICE THAT SET "A" CONTAINS "A," B, C, D, E, AND SET C
ALSO CONTAINS, "A," B, C, D, E, BUT IN A DIFFERENT ORDER.
AND SINCE THE ORDER DOESN'T MATTER
WE CAN SAY THAT C IS A SUBSET OF "A," OR "A" IS A SUBSET OF C.
BUT SINCE "A" AND C ARE EQUAL TO EACH OTHER
THEY ARE NOT PROPER SETS OF EACH OTHER.
SO WE CAN SAY THAT C IS NOT A PROPER SET OF "A,"
OR "A" IS NOT A PROPER SUBSET OF C.
LET'S TAKE A LOOK AT TWO MORE EXAMPLES.
WE WANT TO USE EITHER THE SUBSET SYMBOL
OR THE PROPER SUBSET SYMBOL WHEN APPROPRIATE,
GIVEN THESE TWO SITUATIONS.
SO SET "A" IS THE SET OF PEOPLE
LIVING IN PHOENIX, ARIZONA, U.S.A;
AND SET B IS THE SET OF PEOPLE LIVING IN ARIZONA.
AND SINCE PHOENIX, ARIZONA IS IN THE STATE OF ARIZONA,
WE COULD SAY THAT "A" IS A SUBSET OF B.
AND SINCE SET "A" DOES NOT = SET B,
WE CAN SAY THAT "A" IS ALSO A PROPER SUBSET OF B.
AND SINCE BEING A PROPER SUBSET GIVES US MORE INFORMATION
ABOUT THE TWO SETS,
THE PROPER SUBSET NOTATION WOULD BE MORE APPROPRIATE,
EVEN THOUGH THEY ARE BOTH TRUE.
LOOKING AT OUR SECOND EXAMPLE,
C IS THE SET OF WEEKDAYS, AND D IS THE SET MONDAY,
TUESDAY, WEDNESDAY, THURSDAY, AND FRIDAY.
SO THESE TWO SETS ARE ACTUALLY EQUAL TO EACH OTHER,
SO WE CAN SAY THAT C IS A SUBSET OF D,
AND D IS ALSO A SUBSET OF C.
BUT SINCE THEY'RE NOT EQUAL
WE CANNOT USE THE PROPER SUBSET NOTATION.
SO THIS WOULD BE THE BEST NOTATION TO USE
IN THIS SITUATION.
AND THIS MIGHT REMIND YOU OF INEQUALITY SYMBOLS.
REMEMBER WHEN WE WERE WORKING WITH INEQUALITY SYMBOLS,
IF WE HAVE THIS SYMBOL IT'S JUST LESS THAN;
IF WE HAVE THIS SYMBOL HERE IT'S LESS THAN OR EQUAL TO.
SO THIS BAR HERE REPRESENTS THAT THE TWO SETS
COULD BE EQUAL TO EACH OTHER,
WHEREAS -- PROPER SUBSETS CANNOT BE EQUAL TO EACH OTHER.
A FEW MORE MINOR THINGS TO MENTION ABOUT SUBSETS.
THE EMPTY SET IS A SUBSET OF ANY SET.
AND THE EMPTY SET IS A PROPER SET FOR ALL SETS
EXCEPT FOR ITSELF, THE EMPTY SET.
FROM THESE TWO STATEMENTS IT FOLLOWS THAT IF SET "A"
HAS N ELEMENTS, IT HAS 2 TO THE NTH POWER SUBSETS.
AND IF SET "A" HAS N ELEMENTS,
IT HAS 2 TO THE N - 1 PROPER SUBSETS.
NOTICE THE - 1 BECAUSE WE CAN'T COUNT "A" AS ITS OWN SUBSET.
LET'S TAKE A LOOK AT A COUPLE OF EXAMPLES OF THIS.
"DETERMINE THE NUMBER OF SUBSETS
AND PROPER SUBSETS OF THE GIVEN SETS."
SO NUMBER ONE HAS TWO ELEMENTS, SO N = 2.
SO THERE'S SUPPOSED TO BE 2 TO THE NTH POWER SUBSETS;
SO 2 TO THE 2ND WOULD GIVE US FOUR SUBSETS.
AND THE PROPER SUBSETS WITH 2 TO THE 2ND - 1,
THAT'LL BE THREE PROPER SUBSETS.
LET'S GO AHEAD AND LIST THEM FOR THIS EXAMPLE.
SET B WOULD BE A SUBSET OF B CONTAINING THE ELEMENT "A."
SET C WOULD BE A SUBSET IF IT CONTAINED THE ELEMENT B.
SET D WOULD BE A SUBSET IF IT CONTAINED THE ELEMENT AB.
AND THEN LASTLY, E WOULD BE A SUBSET IF IT WAS THE EMPTY SET.
SO NOTICE WE HAVE FOUR SUBSETS.
AND THEN FOR THE PROPER SUBSETS WE'D HAVE TO ELIMINATE SET D
BECAUSE WE COULDN'T USE TWO SETS
THAT ARE EQUAL TO BE PROPER SUBSETS.
FOR NUMBER TWO WE HAVE N = 3,
SO 2 TO THE 3RD WOULD BE THE NUMBER OF SUBSET;
THERE ARE EIGHT SUBSET.
AND FOR THE NUMBER OF PROPER SUBSETS
IT'S GOING TO BE 2 TO THE 3RD - 1;
THAT'LL BE 8 -1 OR 7 PROPER SUBSETS.
IF YOU WANT YOU MAY WANT TO PAUSE THE VIDEO
AND SEE IF YOU CAN LIST ALL OF THEM.
-- WE'RE GOING TO STOP HERE FOR THIS VIDEO.
NEXT WE'LL TAKE A LOOK AT SET OPERATIONS AND VENN DIAGRAMS.
I HOPE YOU FOUND THIS HELPFUL.