- WELCOME TO A LESSON ON VENN DIAGRAMS AND SET OPERATIONS.
THE GOALS OF THE VIDEO ARE TO DEFINE A VENN DIAGRAM,
USE VENN DIAGRAMS TO SHOW RELATIONSHIPS AMONG SETS,
AND ALSO TO DEFINE SET OPERATIONS.
A VENN DIAGRAM IS A VISUAL DIAGRAM
THAT SHOWS THE RELATIONSHIP OF SETS WITH ONE ANOTHER.
THE SET OF ALL ELEMENTS BEING CONSIDERED
IS CALLED THE UNIVERSAL SET,
AS REPRESENTED BY A RECTANGLE AS WE SEE HERE.
THE COMPLEMENT OF "A"
IS THE SET OF ALL ELEMENTS IN U, NOT IN "A."
OR USING SET BUILDER NOTATION,
THE COMPLEMENT OF "A" IS THE SET OF ELEMENTS,
X, SUCH THAT X IS AN ELEMENT OF U,
AND X IS NOT AN ELEMENT OF "A."
SO IF WE LET THE UNIVERSAL SET BE THE INTEGERS
ONE, TWO, THREE, FOUR, AND FIVE,
AND "A" WAS THE SET CONTAINING ELEMENTS ONE, TWO, AND THREE,
THEN THE COMPLEMENT OF "A" WOULD BE THE ELEMENTS IN U
THAT ARE NOT IN "A," SO IT WOULD CONTAIN THE ELEMENTS
FOUR AND FIVE.
SO THE ELEMENTS ONE, TWO, AND THREE WOULD BE INSIDE SET "A,"
AND THE ELEMENTS FOUR AND FIVE WOULD BE OUTSIDE CIRCLE "A."
SET "A" AND B ARE DISJOINT
IF THEY DO NOT SHARE ANY COMMON ELEMENTS.
SO USING A VENN DIAGRAM, SET "A" AND SET B WOULD NOT OVERLAP.
SO HERE IF THE UNIVERSAL SET
WAS A SET CONTAINING ONE, TWO, THREE, FOUR, FIVE, SIX,
AND SET "A" CONTAINED THE ELEMENTS ONE, TWO, THREE,
SET B WOULD NOT BE ABLE TO CONTAIN ONE, TWO, OR THREE,
BUT IT WOULD HAVE TO CONTAIN ELEMENTS
WITHIN THE UNIVERSAL SET.
SO MAYBE B WOULD CONTAIN FOUR AND FIVE.
SO FOR THE VENN DIAGRAM, ONE, TWO, AND THREE
WOULD BE INSIDE SET "A," FOUR AND FIVE WOULD BE INSIDE SET B,
AND SIX WOULD BE OUTSIDE BOTH CIRCLES.
IN THIS DIAGRAM, B IS A PROPER SUBSET OF "A."
THIS MEANS THAT B IS A SUBSET OF "A," BUT B DOES NOT = "A."
USING THE SYMBOLS DISCUSSED IN THE PREVIOUS VIDEO,
WE CAN SAY THAT B IS A SUBSET OF "A," AND B DOES NOT = "A,"
THEREFORE B IS A PROPER SET OF "A."
AND INSTEAD OF USING THIS SYMBOL FOR A PROPER SUBSET,
SOMETIMES YOU'LL SEE THIS SYMBOL HERE USED.
SO IF WE LET THE UNIVERSAL SET CONTAIN THE ELEMENTS
ONE, TWO, THREE, FOUR, FIVE SIX--
AND LET'S SAY SET "A" CONTAINS ONE, TWO, THREE, AND FOUR
SET B WOULD HAVE TO CONTAIN SOME OF THE ELEMENTS SET "A,"
BUT NOT ALL OF THEM BECAUSE THEY'RE NOT EQUAL.
SO MAYBE SET B WOULD CONTAIN ONE AND TWO.
SO WE'D HAVE THE ELEMENTS ONE AND TWO INSIDE --
THAT ARE ALSO INSIDE SET "A."
THREE AND FOUR WOULD BE OUTSIDE SET B BUT INSIDE SET "A."
AND THEN FIVE AND SIX WOULD STILL BE IN THE UNIVERSAL SET.
AND IF WE HAVE EQUAL SUBSETS,
THAT MEANS B IS A SUBSET OF "A," AND BE IS ALSO = A.
SO ALL THE ELEMENTS IN "A" AND B ARE IN THE SAME CIRCLE.
SO THE UNIVERSAL SET WAS ONE, TWO, THREE, FOUR, FIVE.
IF SET "A" CONTAINS ONE, TWO, AND THREE,
THEN IN THIS CASE B ALSO CONTAINS
THE ELEMENTS ONE, TWO, AND THREE.
SO WE'D HAVE ONE, TWO, AND THREE HERE;
AND THEN OUTSIDE "A" AND BE WOULD BE FOUR, FIVE, AND SIX.
SO IF "A" = B, WE CAN SAY THAT "A" IS A SUBSET OF B,
AND B IS ALSO A SUBSET OF "A."
THE INTERSECTION OF "A" AND B
IS THE SET OF ELEMENTS IN BOTH SET "A" AND SET B.
USING SYMBOLS WE HAVE "A"
INTERSECTION B = ALL THE ELEMENTS X,
SUCH THAT X IS CONTAINED WITHIN "A" AND CONTAINED WITHIN B.
SO IF THE UNIVERSAL SET = ONE, TWO, THREE, FOUR, FIVE, SIX,
AND WE HAVE SET "A" = TWO, THREE, AND FOUR,
AND SET B = THREE, FOUR, FIVE,
WE CAN SEE THAT SETS "A" SHARE THE ELEMENTS OF THREE AND FOUR,
SO THAT WOULD BE IN THIS REGION HERE.
SO THREE AND FOUR WOULD BE IN THIS REGION HERE.
AND SET "A" ALSO CONTAINS THE ELEMENT TWO,
SO THERE'D BE A TWO HERE.
AND THEN SET B ALSO CONTAINS THE ELEMENT FIVE,
SO WE'D HAVE A FIVE OUT HERE.
AND THAT LEAVES THE ELEMENTS ONE AND SIX IN THE UNIVERSAL SET.
SO THE INTERSECTION OF "A" AND B
CONTAIN THE ELEMENTS THREE AND FOUR.
AND NOW LET'S TALK ABOUT THE UNION OF SETS.
THE UNION OF "A" AND B IS THE SET OF ALL ELEMENTS
IN SET "A" OR SET B.
SO "A" UNION B = ALL THE ELEMENTS X,
SUCH THAT X IS CONTAINED WITHIN "A" OR X IS CONTAINED WITHIN B.
SO THE BIG DIFFERENCE BETWEEN INTERSECTION
AND UNION IS INTERSECTION
WE HAVE AN AND, UNION WE HAVE AN OR.
USING THE SAME SETS AS WE DID FOR THE INTERSECTION EXAMPLE,
IT WOULD LOOK THE SAME EXCEPT
NOW THE UNION WOULD BE ALL THE ELEMENTS THAT ARE IN "A" OR B.
SO WE'D BE TALKING ABOUT THIS ENTIRE REGION HERE,
BECAUSE AN ELEMENT ONLY HAS TO BE IN ONE OF THEM
IN ORDER TO BE IN THE UNION.
SO "A" UNION B IN THIS CASE WOULD BE ALL THE ELEMENTS
IN THE UNIVERSAL SET EXCEPT ONE AND SIX.
SO WE'D HAVE TWO, THREE, FOUR, AND FIVE.
AND A COUPLE THINGS TO MENTION ABOUT THE EMPTY SET--
ANY SET "A" INTERSECTED WITH THE EMPTY SET = THE EMPTY SET.
AND THE UNION OF ANY SET "A"
AND THE EMPTY SET WILL ALWAYS BE THE SET "A."
I THINK WE'LL STOP HERE FOR PART ONE
AND TAKE A LOOK AT SOME ADDITIONAL EXAMPLES IN PART TWO.