The Language of Sets MATH 102 Contemporary Math S. Rook

Overview Section 2.1 in the textbook: – Representing sets – More with sets

Representing Sets

4 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

5 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. is pronounced “is an element of the set of”

6 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?

7 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}

8 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}

Converting Between Roster & Set- Builder Notation (Example) Ex 1: Convert to roster notation: a) b) A = {y | y is a letter in the word music} 9

Converting Between Roster & Set- Builder 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} 10

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 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)A = Ø b)B = { …, -10, -9, -8, -7, -6} c)C = { {2, 3, 5}, {7, 11}, {13, 17, 19} } d) D = {d | d is a person in this class today} e)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)