Transcript Set-Builder Notation

The Language of Sets
MATH 102
Contemporary Math
S. Rook
Overview
• Section 2.1 in the textbook:
– Representing sets
– More with sets
Representing Sets
Sets
• Set: a collection of objects. Each object is
known as a member or an element of the set
– We usually use capital letters to denote sets and
lowercase letters to denote elements of sets
• Two ways to express sets:
– Set-Builder notation
– Roster notation
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Set-Builder Notation
• Set-Builder Notation: describes, but does not
explicitly list the elements of a set
e.g. {x | x is an even number},
• The | (vertical bar) is pronounced “such that”
e.g. x | x  integers
, x  0
•  is pronounced “is an element of the set of”
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Roster Notation
• Roster Notation means to explicitly list the
elements of a set
– When listing elements, we use set notation and
place the elements between and left { and right }
(called curly braces)
– We use … (ellipses) to denote a set extending
infinitely in the same pattern
• The set of even numbers can then be expressed as {0,
2, 4, 6, …}
• How can we express {x | x ε odd number} in roster
notation?
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Common Sets of Numbers
• Sets of numbers to be familiar with:
– Natural numbers (counting numbers):
{1, 2, 3, 4, 5,…}
{x | x ε N}
– Whole numbers: the natural numbers along with 0.
{0, 1, 2, 3, 4,…}
{x | x ε W}
– Integers: the natural numbers, opposite of the
natural numbers, and zero.
{…, -2, -1, 0, 1, 2,…}
{x | x ε I}
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Common Sets of Numbers (Continued)
– Rational numbers: any number that can be
expressed as the quotient of two integers, a, b,
b ≠ 0.
{a⁄b | a and b are integers, b ≠ 0}
{x | x ε Q}
– Real numbers: any number that lies on the
number line
{x | x ε R}
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Converting Between Roster & SetBuilder Notation (Example)
Ex 1: Convert to roster notation:
a)
B  x | x  integers , x   3 AND x  1
b) A = {y | y is a letter in the word music}
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Converting Between Roster & SetBuilder Notation (Example)
Ex 2: Convert to set-builder notation:
a) C = {-5, -4, -3, -2, -1}
b) D = {…, -2, -1, 0, 1, 2, 3, 4}
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More with Sets
Null/Empty Set
• Null/Empty Set: a set that contains NO
elements
– Represented as { } or Ø
– What would be an example of an empty set
written in set-builder notation?
• Does { } have same meaning as {Ø}?
– Think of the curly braces as a container/bag
– See pg 41 in the textbook
Set Membership
• Recall that we use the  symbol to denote an
element that can be found within a set
• MUST be careful on precise usage:
–
–
–
–
–
–
–
Is the statement 1 ε {1, 2, 3} true?
Is the statement {1} ε {1, 2, 3} true?
Is the statement {2} ε { {1}, {2}, {3} } true?
Is the statement {2} ε { {1, 2}, {3} } true?
Is the statement 1 ε { {1}, 2, 3} true?
Is the statement Ø ε {Ø} true?
Is the statement Ø ε {1, 2, 3} true?
Set Membership (Example)
Ex 3: Replace # with  or  to make the statement
true:
a) 3 # {x | x is a whole number}
b) Tiger Woods # {a | a is a professional ice
skater}
c) {1, 5} # { {1}, {5}, {1, 5} }
d) 0 # Ø
e) Ø # {0}
Cardinal Number of a Set
• Cardinal number: the number of elements in
a set A denoted by n(A)
– Sets that have a countable number of elements
are called finite
– Sets that have a number of elements that are not
countable are infinite
Cardinal Number of a Set (Example)
Ex 4: Identify the set as finite or infinite. If
finite, list the cardinal number of the set.
a)
b)
c)
d)
e)
A=Ø
B = { …, -10, -9, -8, -7, -6}
C = { {2, 3, 5}, {7, 11}, {13, 17, 19} }
D = {d | d is a person in this class today}
E = {e | e is a digit in π}
Summary
• After studying these slides, you should know how to do
the following:
– Express the elements of a set in set-builder or roster
notation
– Understand the concept of the null/empty set
– State whether a given element is a member of a set
– Classify sets as finite or infinite
– Find the cardinal number of a finite set
• Additional Practice:
– See the list of suggested problems for 2.1
• Next Lesson:
– Comparing Sets (Section 2.2)