Download Section 1.1 Introduction to Sets Set: A set is a collection of objects

Section 1.1 Introduction to Sets
Set: A set is a collection of objects. Each object in a set is called an element or member of the
set. A set is usually given a capital letter name.
Example: Let A be the set of integers between 2 and 5 inclusive. List the elements of set A
and place the elements in set braces.
Note: This question wouldn’t make sense to ask unless A is a set. The bar is set quite low to
be a set. To be a set, A needs collection of objects (elements). There is no requirement on
what the elements need to be. The elements may be numbers, symbols, letters and words.
Really, they can be anything.
A is a collection of numbers (in this case the elements are numbers). Thus A is a set.
There are two words that are bolded in the description of set A that we need to understand
in order to list the elements that make up set A. The bolded words are: integers and
inclusive.
Integers are essentially numbers without decimals, whether positive or negative.
Integers = {…,-5,-4,-3,-2,-1,0,1,2,3,4,5…}
Inclusive means to include the first and the last number of the numbers that are described.
Thus set A contains numbers without decimals between 2 and 5 and I need to include the 2
and the 5.
Answer: A = {2,3,4,5} (I wrote the A to the left of an equal sign as it is the name of this set)
The curly braces in this definition are called set braces, and they are common to use when
listing the elements (members) of a set.
The order that the elements that set A are listed in is not important and has no effect on the set
itself. The set A described above can have its elements listed many ways. These are two more
ways to list the elements of the set A. These would both be correct answers to the example.
A = {4,2,3,5}
A = {5,4,3,2}
Had my example used the words exclusive instead of inclusive the 2 and the 5 would NOT
be included. When the word exclusive is used in a set description the first and the last
number of the given range are not included when we list the elements of the set.
Example: Let A be the set of integers between 2 and 5 exclusive. List the elements of set A
and place the elements in set braces.
Answer: A = {3,4}
Here is another example that will help with the homework below.
Example: Let B be the set of solutions to the problem: x + 5 = 8. List the elements of set B
and place the elements in set braces.
I need to do a bit of Algebra to find the elements of set B.
x+5=8
__-5_-5
x=3
3 is the only element in set B, as it is the only solution to the problem that defines set B.
Answer: B = {3}
Homework #1 – 8. List all of the elements in the sets described below. Write your answer
using set braces.
1) The set A of even numbers between 1 and 10 inclusive.
2) The set A of odd numbers between 1 and 10 inclusive.
3) The set B of even numbers between 1 and 10 exclusive.
4) The set B of odd numbers between 1 and 10 exclusive.
5) The set A of solutions to the problem: x + 6 = 10
6) The set A of solutions to the problem x – 4 = 9
7) The set B of flavors of ice cream in Neapolitan ice cream.
8) The set B of days of the week.
A set should be well-defined.
A set is well-defined if any informed objective person can decide if a given element is in the
set or not.
This definition is a bit abstract. Usually a set defined with words such as good, bad, nice are
indicators that the set is not well defined.
Example: Let M be the set of months with more than 30 days. Is the set well-defined?
This is well defined as there is no ambiguity as to whether a month belongs in the set or
not.
This set contains the months: January, March, May, July, August, October and December.
Answer: The set is well-defined.
Example: Let C be the set of good cell phones. Is this set well-defined?
This is not well-defined as whether a cell phone is good or not likely depends on who is
asked.
Answer: The set is not well-defined.
Homework #9 – 16. Which of the following sets are well-defined? Write yes for your answer if
the set is well-defined. Write no for your answer if the set is not well-defined.
9) A is the set of goofy dogs.
10) B is the set of GCC students whose GPA is 3.0 or greater.
11) D is the set of numbers whose square is 16.
12) C is the set of good GCC students.
13) The set of years that the Arizona Cardinals won their division.
14) The set of Great Lakes.
15) The set of cities that have nice climates.
16) The set of good smart phones
Finite Sets versus Infinite Sets
A set is finite if it is possible, given enough time, to write down every element in the set.
A set is infinite if it is not finite.
Example: Consider the set A = {1, 2, 3, …, 20} Is A finite or infinite?
(Note: The three periods in this definition are called ellipses and mean that you should
continue with the established pattern.)
A can be written as: A = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
I was able to write down every element of set A, thus set A is finite.
Answer: Finite
Example: Consider the set B = {1, 2, 3, …}. Is B finite or infinite?
B = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18…}
No matter how much I write there is still more of B. It is impossible to write all of the elements
of set B, hence set B is infinite.
Answer: Infinite
Example: Determine whether the set is infinite or finite. The set A is the set that contains the
factors of 10.
We need to know what it means to be a factor of 10 to answer this question. Factors are
numbers that divide evenly into a number.
The factors of 10 are 1, 2, 5, 10,-1,-2,-5,-10.
I am able to write down all of the elements of A, so A is finite.
Answer: Finite
Example: Determine whether the set is infinite or finite. The set of multiples of 4.
We need to know what it means to be a multiple of 4 to answer this question. The multiples of
4 are any number that 4 divides evenly into
Here are the positive multiples of 4: {4,8,12,16,20…}
It is impossible to list all of the multiples of 4, so the set is infinite.
Answer: Infinite
Homework #17 – 24. Determine whether each set is finite or infinite.
17) The set of factors of 12.
18) The set of factors of 24.
19) The set of multiples of 3.
20) The set of multiples of 5.
21) The set of positive multiples of 4 that are less than 30.
22) The set of multiples of 4 that are greater than 10 and less than 50.
23) The set of even positive integers.
24) The set of odd positive integers.
The empty set is something that I will refer to from time to time.
The set containing no elements is called the empty set, or null set. The empty set is
denoted by { }, or by the small Greek letter phi, ∅.
This set is an example of an empty set: The set A of the months of the year in which the
average high temperature in Phoenix is less than 10 degrees Fahrenheit.
A can be written many ways. It is set with no elements. We may write:
A = { } or
A=∅
or
A = empty set
More Notation: The symbol ∈ stands for the word element. An object is an element of a set if
it is a list of all of the elements of a particular set.
5 ∈ A means “5 is an element of the set A.”
When you see the notation 5 ∈ A, it means that 5 will be in the list if you make a list of all of
the elements of set A.
6∉A means “6 is not an element of set A”
When you see the notation 6∉A, it means that 6 will NOT be in the list if you make a list of all of
the elements of A.
Example: Consider the following statements with regard to set B = {1,2,3,4,5}. State whether
the statement is true or false.
a) 1 ∈ B
Answer: true, because 1 is in the list of all of the elements of set B
b) 1∉B
Answer: False, for this to be true 1 could not be in the list of the elements of set B.
c) 8∉B
Answer: true, because 8 is not in the list of all of the elements of set B
d) 8 ∈ B
Answer: false, for this to be true 8 would need to be in the list of elements of set B.
The set of Natural numbers is a set you should be familiar with. The set of Natural numbers is
often assigned the letter N, and it contains integers greater than 0.
N = set of Natural numbers = {1,2,3,4,5…}
The Cardinal Number of a finite set A is the number of elements of A.
The cardinal number of a set A is denoted n(A).
Example: Find the cardinal number of set A. A = {2,3,4,5,6,7}
There are 6 elements in set A. Thus the cardinal number of set A is 6.
Answer: n(A) = 6
Example: Find n(A) where A is the set of natural numbers less than 8.
I first need to list the elements of set A. Note, 8 is not an element of set A. 8 would be an
element of set A had the problem read: A is the set of natural numbers less than or equal to 8.
A key word in the definition of set A is LESS THAN. The word less than is similar to the word
exclusive. The end point of the set is not included when we list the elements of the set.
A contains the elements {1,2,3,4,5,6,7}.
A has 7 elements.
Answer: n(A) = 7
Homework #25 – 34. Find the cardinal number of the following sets. That is find the number of
elements in each set.
25) The set B of natural numbers between 5 and 10 inclusive.
26) The set B of even natural numbers between 5 and 10 inclusive.
27) The set A of even natural numbers greater than 4 and less than 11.
28) The set A of odd natural numbers greater than 2 and less than or equal to 5.
29) The set A of natural numbers less than 9.
30) The set A of natural numbers less than or equal to 5.
31) Find n(A) where A = {1,3,5}
32) Find n(B) where B = {a, c, e, o, v}
33) Find n(C) where C = {2,4,6,8,10}
34) Find n(D) where D = {0}
Definition of Set Equality
Two sets are equal if they contain precisely the same elements.
Example: Are sets A and B equal? A = {a,b,c,d,e} and B = {e,d,c,a,b}
Answer: yes (since they have exactly the same elements.)
Example: Are sets A and B equal? A = {1,2,3} and B = {a,b,c}
Answer: no, (since they do not have exactly the same elements.)
Example: Are sets A and B equal? A = {1,2,3} and B = {1,2,3,4}
Answer: no (A doesn’t have a 4 which B has so they don’t have exactly the same elements so
they are not equal.)
Set Equivalence for Finite Sets
Finite set A is equivalent to set B if n(A) = n(B). That is two sets are equivalent if they have the
same number of elements.
Example: Are sets A and B equivalent? A = {a,b,c,d,e} and B = {e,d,c,a,b}
Answer: yes (since they have exactly the same NUMBER of elements. They both have 5
elements)
Example: Are sets A and B equivalent? Where A = {1,2,3} and B = {a,b,c}
Answer: yes, (since they have exactly the same NUMBER of elements. They both have 3
elements.)
Example: Are sets A and B equivalent? Where A = {1,2,3} and B = {1,2,3,4}
Answer: no (A has 3 elements and B has 4 elements.) To be equivalent they need to have the
same number of elements.
Example: Determine whether the sets A and B are equal, equivalent, neither.
A = {a,b,c,d} B= {a,e,i,o}
These sets have the same number of elements, so they are equivalent.
They don’t have exactly the same elements so they are not equal.
Answer: only equivalent
Example: Determine whether the sets A and B are equal, equivalent, neither.
A = {a,b,c,d} B= {a,c,d,b}
These sets have the same number of elements, so they are equivalent.
They have exactly the same elements so they are equal.
Answer: both equal and equivalent
Example: Determine whether the sets A and B are equal, equivalent, neither.
A = {b,c,d} B= {a,c,d,b}
These sets do not have the same number of elements, so they are not equivalent.
They do not have exactly the same elements so they are not equal.
Answer: neither
Homework #35– 45. Determine whether the sets A and B are equal, equivalent, neither.
35) A = {d,a,b,c} B = {a,b,c,d}
36) A = {1,2,3,4} B = {4,3,2}
37) A = {a,b} B = {c,b,a}
38) A = {a,b} B = {b, a}
39) A = {0,1,2,3,4} B = {a,b,c,d,e}
40) A = {1,3,2} B = {lemon, cherry, apple}
41) A = {2,1} B = {1,4}
42) A = {1,2,3} B = {4,5,6,7}
43) A = {1} B = {2, 1}
44) A = {1,2,3,4,5} B = {4,5,3,2,1}
45) A = {1,3,5,7} B = {7,3,5,1}
Answers: 1) A = {2,4,6,8,10} 3) B = {2,4,6,8} 5) A = {4} 7) B = {chocolate, vanilla, strawberry}
9) no 11) yes 13) yes 15) no 17) finite 19) infinite 21) finite 23) infinite
25) n(B) = 6 27) n(A) = 3 29) n(A) = 8 31) n(A) = 3 33) n(C) = 5 35) Equal and equivalent
37) neither 39) equivalent 41) equivalent 43) neither 45) equal and equivalent