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- WELCOME TO A LESSON ON SET THEORY BASICS.
THE GOALS OF THIS VIDEO ARE TO DEFINE BASIC SET VOCABULARY
AND SET CONCEPTS.
A SET IS A COLLECTION OF DISTINCT OBJECTS,
AND THE OBJECTS CAN BE CALLED ELEMENTS
OR MEMBERS OF THE SET.
A SET DOES NOT LIST AN ELEMENT MORE THAN ONCE
SINCE AN ELEMENT IS EITHER A MEMBER OF A SET
OR IT'S NOT A MEMBER.
WE'VE ACTUALLY ALREADY STUDIED SETS WHEN WE DEFINED
THE SET OF REAL NUMBERS.
REMEMBER, ALL REAL NUMBERS ARE MADE UP OF EITHER RATIONAL
OF IRRATIONAL NUMBERS.
AND THEN WE CAN BREAK THOSE SETS OF NUMBERS
INTO MORE SPECIFIC SETS OF NUMBERS AS WELL.
AND CAPITAL LETTERS ARE NORMALLY USED TO IDENTIFY A SET.
NOW, THERE ARE SETS THAT HAVE REPEATED ELEMENTS
AND THESE ARE CALLED MULTI SETS.
AND THIS WILL NOT BE ADDRESSED IN THIS VIDEO.
SO, LOOKING AT OUR SETS OF REAL NUMBERS AGAIN,
NOTICE THAT ALL REAL NUMBERS ARE MADE UP OF RATIONAL NUMBERS
AND IRRATIONAL NUMBERS.
SO WE COULD CALL RATIONAL AND IRRATIONAL NUMBERS
A SUBSET OF REAL NUMBERS, BUT WE'LL TALK ABOUT SUBSETS
IN THE NEXT VIDEO.
RIGHT NOW, LET'S GO AHEAD AND FOCUS ON THE DIFFERENT WAYS
OF IDENTIFYING A SET.
THE FIRST WAY IS A WRITTEN DESCRIPTION.
WE COULD STATE THAT SET "A" IS THE SET OF CALENDAR MONTHS
BEGINNING WITH THE LETTER J.
WE COULD ALSO IDENTIFY SET "A" USING THE LIST
OR ROSTER METHOD WHERE WE LIST THE ACTUAL MEMBERS OF THE SET.
SO SET "A" IS EQUAL TO JANUARY, JUNE, AND JULY.
OR WE COULD ALSO USE SOMETHING CALLED "SET BUILDER NOTATION."
SET "A" IS THE SET OF ALL ELEMENTS X
SUCH THAT X IS A CALENDAR MONTH BEGINNING WITH THE LETTER J.
ALL OF THESE REPRESENT THE SAME SET.
SO, TO DESCRIBE THE SET OF THESE THREE COLORS,
THE WRITTEN DESCRIPTION COULD BE "B IS THE SET...
OF PRIMARY COLORS."
OR USING THE ROSTER METHOD, WE COULD SAY THAT B
IS EQUAL TO THE SET WHERE THE COLORS WOULD BE
RED, BLUE, AND YELLOW.
OR USING THE SET BUILDER NOTATION,
SET B WOULD BE EQUAL TO THE SET OF X
SUCH THAT X IS A PRIMARY COLOR.
THERE'S ALSO A SPECIAL SET CALLED THE "EMPTY SET"
OR "NULL SET."
THIS IS A SET HAVING NO ELEMENTS AT ALL.
THE NOTATION USED FOR AN EMPTY SET
WOULD BE EITHER OF THESE TWO NOTATIONS.
FOR EXAMPLE, IF B IS THE SET OF 5-SIDED SQUARES,
SINCE THERE AREN'T ANY 5-SIDED SQUARES,
B WOULD BE THE NULL SET, OR EMPTY SET,
AS IDENTIFIED HERE.
WE CAN ALSO USE THESE TWO SYMBOLS HERE
TO INDICATE IF AN ELEMENT IS A MEMBER OF A GIVEN SET.
SO FOR EXAMPLE, IF WE HAVE SET C IS THE SET OF EVEN NUMBERS
GREATER THAN 2 AND LESS THAN 12, LET'S FIRST IDENTIFY THIS SET
USING THE ROSTER METHOD.
SO WE CAN'T INCLUDE 2 BECAUSE 2 IS NOT GREATER THAN 2.
SO WE'D HAVE 4, 6, 8, 10.
AND WE WOULD ALSO NOT INCLUDE 12 BECAUSE 12 IS NOT LESS THAN 12.
SO THIS IS OUR SET.
SO WE CAN SAY THAT 4 IS AN ELEMENT OF SET C,
6 IS AN ELEMENT OF SET C, 8 IS AN ELEMENT OF SET C,
AND SO IS 10.
BUT WE COULD ALSO SAY THAT 2 IS NOT AN ELEMENT OF SET C,
AND 12 IS NOT AN ELEMENT OF SET C.
AND OF COURSE, THERE ARE MANY OTHERS.
NOW LET'S TALK ABOUT CARDINALITY.
THE CARDINALITY OR CARDINAL NUMBER OF A SET
IS THE NUMBER OF ELEMENTS IN THE SET.
AND WE NORMALLY USE THESE TWO NOTATIONS TO REPRESENT
THE CARDINALITY OF A SET.
AND SINCE SET NOTATION DOES NOT REPEAT ELEMENTS,
WE JUST HAVE TO COUNT THE NUMBER OF ELEMENTS IN EACH SET
TO IDENTIFY THE CARDINALITY OF THE SET.
SO THE CARDINALITY OF SET "A" WOULD JUST BE 1, 2, 3, 4, 5, 6
BECAUSE THERE ARE SIX ELEMENTS IN THE SET.
THE CARDINALITY OF SET B WOULD BE
1, 2, 3, 4, 5, 6 AS WELL.
AND THE CARDINALITY OF SET C WOULD BE 3.
NOW, TWO SETS ARE EQUIVALENT IF THEY CONTAIN
THE SAME NUMBER OF ELEMENTS.
IF TWO SETS ARE EQUIVALENT, THEY CAN BE PLACED
IN A 1 TO 1 CORRESPONDENCE.
SO IF TWO SETS ARE EQUIVALENT, WE WRITE THEY'RE EQUIVALENT
USING THIS NOTATION HERE.
SO LOOKING AT THESE TWO SETS HERE,
WE COULD SAY THAT SET "A" IS EQUIVALENT TO SET B
BECAUSE THEY HAVE THE SAME CARDINALITY
OR THE SAME NUMBER OF ELEMENTS.
NOW, WE HAVE TO BE CAREFUL ABOUT THIS WORD "EQUIVALENT."
OFTEN IN MATH IT'S USED WHEN THINGS ARE EQUAL.
BUT OBVIOUSLY THESE SETS ARE NOT EQUAL,
THEY JUST HAVE THE SAME NUMBER OF ELEMENTS.
NOW, THIS IDEA OF A 1 TO 1 CORRESPONDENCE
JUST MEANS THAT FOR EVERY ONE ELEMENT
IN SET "A," WE CAN MATCH IT WITH ONE ELEMENT IN SET B.
SO WE COULD MATCH 1 WITH 2, 2 WITH 3, 3 WITH 8,
5 WITH 11, 8 WITH 12, AND 9 WITH 13,
ILLUSTRATING THE 1 TO 1 CORRESPONDENCE.
THE LAST TOPIC WE'LL TALK ABOUT IN THIS VIDEO
IS WHEN TWO SETS ARE EQUAL.
TWO SETS ARE EQUAL IF THEY CONTAIN
THE EXACT SAME ELEMENTS THAT MAY OR MAY NOT BE
IN THE SAME ORDER.
SO, LOOKING AT THESE THREE SETS HERE,
NOTICE THAT SET "A" AND SET C HAVE THE SAME CARDINALITY,
MEANING THEY HAVE THE SAME NUMBER OF ELEMENTS
AND THEY ALSO HAVE THE EXACT SAME ELEMENTS.
SET "A" AND C CONTAIN LITTLE "A."
SET "A" AND C CONTAIN LITTLE B.
SET "A" AND C CONTAIN LITTLE C,
AND SET "A" AND C CONTAIN LITTLE D.
SO THEY HAVE THE EXACT SAME ELEMENTS, THEREFORE,
THE TWO SETS ARE EQUAL.
SET "A" IS EQUAL TO SET C.
THE LAST THING I WANT TO MENTION IS THAT ALL OF THE SETS
WE HAVE SEEN IN THIS VIDEO HAVE BEEN FINITE SETS,
MEANING THEY HAVE A CERTAIN NUMBER OF ELEMENTS
OR TERMS.
THE CARDINALITY OF THE EMPTY SET IS 0
BECAUSE THE EMPTY SET HAS 0 ELEMENTS.
BUT THE CARDINALITY OF AN INFINITE SET
WE SAY IS "UNCOUNTABLE."
HOWEVER, WE CAN USE SOMETHING CALLED
"TRANSFINITE CARDINAL NUMBERS" TO DESCRIBE THE SIZE
OF AN INFINITE SET.
BUT RIGHT NOW WE'RE JUST GOING TO IDENTIFY THE CARDINALITY
OF AN INFINITE SET AS "UNCOUNTABLE."
I HOPE YOU FOUND THIS INTRODUCTORY VIDEO HELPFUL.
IN THE NEXT VIDEO WE'LL TALK ABOUT SUBSETS.
THANK YOU FOR WATCHING.