- WELCOME TO A LESSON ON SET BUILDER NOTATION.
IN THIS LESSON, WE WILL DEFINE SET BUILDER NOTATION,
REPRESENT A SET USING SET BUILDER NOTATION,
AND THEN ALSO COMPARE SET BUILDER NOTATION
TO OTHER WAYS TO REPRESENT SETS OF NUMBERS.
A COLLECTION OF NUMBERS CAN BE DESCRIBED AS A SET.
FOR EXAMPLE, THIS IS A SET CONTAINING THE NUMBERS
1, 3, 5, 9, AND 13.
THESE NUMBERS ARE ALSO OFTEN CALLED ELEMENTS.
SET BUILDER NOTATION CAN BE USED TO BUILD OR DESCRIBE A SET.
THIS IS ESPECIALLY HELPFUL IF THE SET HAS INFINITE NUMBER
OF NUMBERS OR AN INFINITE NUMBER OF ELEMENTS.
SO LOOKING AT AN EXAMPLE, THIS IS SET BUILDER NOTATION.
WE READ THESE AS THE SET OF ALL "X"s SUCH THAT
"X" IS GREATER THAN OR EQUAL TO -2.
NOTICE IN THE SECOND CASE, WE'VE REPLACED A COLON
WITH A VERTICAL BAR.
THESE ARE THE TWO MOST COMMON WAYS
TO REPRESENT SET BUILDER NOTATION,
AND THESE TWO ARE EQUIVALENT.
LET'S TAKE A LOOK AT A COUPLE MORE EXAMPLES,
BEFORE WE TRY IT OURSELVES.
HERE, WE HAVE THE SET OF NUMBERS SATISFYING THE FORMULA
2X + 1 SUCH THAT "X" IS AN INTEGER.
THIS MEANS "X" IS AN ELEMENT OF THE SET OF INTEGERS.
SO IF "X" IS AN INTEGER, THEN WE SUBSTITUTE THOSE VALUES
IN TO THE FORMULA 2X + 1.
THIS WOULD GIVE US A SET OF ODD INTEGERS.
NEXT, WE HAVE THE SET OF ALL "X"s
THAT ARE REAL NUMBERS SUCH THAT "X" = X SQUARED.
SO THE ONLY REAL VALUES THAT SATISFY "X" = X SQUARED
WOULD BE WHEN X IS ZERO AND X IS ONE.
SO THIS SET BUILDER NOTATION REPRESENTS THE SET
CONTAINING ZERO AND ONE.
AND THEN FINALLY WE HAVE THE SET OF ALL "X"s
SUCH THAT "X" DOESN'T EQUAL ZERO.
THIS WOULD BE THE SET OF ALL REAL NUMBERS,
EXCEPT ZERO.
SO IN GENERAL WE USE THE FOLLOWING STRUCTURE
WHEN USING SET BUILDER NOTATION.
WE START WITH THE FORMULA FOR THE ELEMENTS
AND THEN A VERTICAL BAR OR COLON FOLLOWED
BY THE RESTRICTIONS.
NOW LET'S TAKE A LOOK AT OUR OWN EXAMPLES.
WE'RE ASKED TO GRAPH THE GIVEN INTERVAL
THAT EXPRESS USING INEQUALITIES AND SET BUILDER NOTATION.
SO THIS IS A NICE REVIEW OF INTERVAL NOTATION,
INEQUALITIES, SET BUILDER NOTATION,
AS WELL AS GRAPHING AN INTERVAL ON THE NUMBER LINE.
SO WE'LL BEGIN BY GRAPHING THIS INTERVAL
FROM NEGATIVE INTERVAL TO THREE WHERE THE INTERVAL
IS CLOSED ON THREE.
REMEMBER THE SQUARE BRACKET MEANS THREE IS INCLUDED
IN THE INTERVAL, SO WE'LL HAVE
A CLOSED POINT ON POSITIVE THREE.
SO IF THIS IS OUR NUMBER LINE,
LET'S SAY THIS IS ZERO AND THIS IS THREE,
THE INTERVAL INCLUDES +3,
SO WE HAVE A CLOSED POINT HERE.
AND WE HAVE AN ARROW TO THE LEFT APPROACHING NEGATIVE INFINITY.
SO NOW USING INEQUALITY, WE CAN SAY
THAT "X" IS LESS THAN OR EQUAL TO +3 AND THEREFORE,
USING SET BUILDER NOTATION, WE CAN NOW SAY THE SET
OF ALL "X"s SUCH THAT "X" IS LESS THAN OR EQUAL TO THREE.
SO MANY CASES YOU CAN JUST VIEW SET BUILDER NOTATION
AS A MORE FORMAL WAY TO REPRESENT AN INTERVAL,
RATHER THAN JUST USING AN INEQUALITY.
LET'S TAKE A LOOK AT ANOTHER EXAMPLE.
IN THIS EXAMPLE, WE WANT TO GRAPH
THE INTERVAL THEN EXPRESS, USING INTERVAL NOTATION
AND SET BUILDER NOTATION.
NOTICE HERE, OUR SET IS GIVEN USING A COMPOUND INEQUALITY.
SO BEGIN BY GRAPHING THIS INTERVAL.
LET'S SAY THIS IS ZERO, AND THIS IS -2.
AND THIS IS 6.
THIS IS THE INTERVAL WHERE "X" IS GREATER THAN -2
AND LESS THAN OR EQUAL TO 6.
SO THIS INTERVAL DOES NOT INCLUDE -2,
BUT IT DOES INCLUDE +6.
SO WE'D HAVE AN OPEN POINT ON - 2, A CLOSED POINT ON +6,
AND THE INTERVAL BETWEEN.
SO THERE'S THE GRAPH OF OUR INTERVAL.
NOW WE WANT TO EXPRESS THIS USING INTERVAL NOTATION.
SO THE INTERVAL STARTS AT -2 AND ENDS AT +6,
BUT IT DOES NOT INCLUDE -2.
SO WE USE A ROUNDED PARENTHESIS HERE.
IT DOES INCLUDE 6, SO WE USE A SQUARE BRACKET HERE.
AND THEN FINALLY FOR SET BUILDER NOTATION,
WE WOULD HAVE THE SET OF ALL "X"s SUCH THAT
"X" IS GREATER THAN -2 AND LESS THAN OR EQUAL TO 6.
WE COULD ALSO EXPRESS THIS A SLIGHTLY DIFFERENT WAY
USING TWO INEQUALITIES CONNECTED WITH AND FOR EXAMPLE,
WE COULD SAY THE SET OF ALL "X"s SUCH THAT "X" IS GREATER THAN -2
AND "X" IS LESS THAN OR EQUAL TO 6.
THESE TWO WOULD BE EQUIVALENT USING SET BUILDER NOTATION.
IN THIS EXAMPLE, WE'RE GIVEN THE GRAPH
OF AN INTERVAL AND ASKED TO STATE THE INTERVAL
USING INTERVAL NOTATION, INEQUALITIES, AND, ONCE AGAIN,
SET BUILDER NOTATION.
NOTICE HOW WE HAVE AN OPEN POINT ON -2, AND THEN THE GRAPH
MOVES TO THE RIGHT APPROACHING POSITIVE INFINITY.
AND THEREFORE, USING INTERVAL NOTATION,
WE'D HAVE THE OPEN INTERVAL FROM -2 TO POSITIVE INFINITY.
NOTICE HOW IT'S A ROUNDED PARENTHESIS HERE BECAUSE IT
DOES NOT INCLUDE -2, AND WE ALWAYS HAVE
A ROUNDED PARENTHESIS FOR INFINITY
AND NEGATIVE INFINITY.
NEXT, FOR INEQUALITY, WE WOULD SAY THAT
"X" IS GREATER THAN -2 AND THEREFORE,
USING SET BUILDER NOTATION, WE WOULD HAVE THE SET
OF ALL "X"s SUCH THAT "X" IS GREATER THAN -2.
SO WHAT YOU'LL RECOGNIZE IN MANY CASES
IS THAT SET BUILDER NOTATION IS JUST MORE OF A FORMAL WAY
TO REPRESENT AN INTERVAL THAN USING
JUST BASIC INEQUALITY NOTATION.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
THIS ONE, WE JUST WANT TO GIVE THE SET BUILDER NOTATION.
FOR THE SET OF NUMBERS LISTED HERE,
WE HAVE THE SET OF EVEN NUMBERS.
ALL EVEN NUMBERS ARE MULTIPLES OF TWO, AND THEREFORE,
USING SET BUILDER NOTATION, WE CAN SAY THAT ALL EVEN NUMBERS
WOULD BE THE SET SATISFYING THE FORMULA TWO TIMES "X"
SUCH THAT "X" WOULD BE AN INTEGER,
WHICH WE CAN SAY "X" IS AN ELEMENT
OF THE SET OF INTEGERS, WHICH WE CAN REPRESENT
USING A CAPITAL "Z."
'COURSE WE COULD ALSO SPELL THIS OUT AND SAY "X" IS AN INTEGER.
I HOPE YOU FOUND THIS LESSON HELPFUL.