Download 7-1 Intro to Sets

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IB Math Studies Year 1
Date_______________
7-1 Intro to Sets
Learning Goal:
#1 What notation do we use for number sets? How do we determine if a given number is a member of a
number system?
Terminology
*Set
*Element
Notation
Symbol
Meaning
n(A)
∈
∈
∅ or { }
Commonly used sets
Prime Numbers
Composite Numbers
Even Numbers
Odd Numbers
Let's take a look together!
1) A is the set of all odd square numbers less than 100.
a. Write down the elements of A.
b. Find n(A)
Try these similar examples!
2) B is the set of all odd numbers between 2 and 12.
a. Write down the elements of B.
b. Find n(B).
3) C is the set of all prime numbers less than 10.
a) Write down the elements of C.
b) Find n(c)
Number Systems
Symbol
Name
List {…} or Examples
ℕ
The Set of
Natural Numbers
{0,1, 2, 3, …
ℤ
The Set of
Integers
ℚ
The Set of
Rational
Numbers
ℝ
The Set of Real
Numbers
ℚ′
The Set of
Irrational
Numbers
Describe in words
{… , −3, −2, −1, 0, 1, 2, 3, … }
All numbers that can be expressed as
𝑎
quotient/fraction of two integers: 𝑏
ℝ = {𝑥| − ∞ < 𝑥 < ∞}
The union of all rational and
irrational numbers; ALL numbers
that can be found on a number line.
4) Determine if each of the given statements is true or false.
a) All rational numbers are real numbers.
b) All integers are natural numbers.
c) All natural numbers are integers.
d) A number cannot be rational and irrational at the same time.
e) An irrational number is not a real number.
Let’s use these number systems in an example!
5) Check off each number system that the given value is a member of.
Natural
Integer
Rational
Irrational Real
1
-2.5
4/2
9.0
¾
𝜋
Set Notation Practice
6) List the elements in the set {𝑥 ∈ ℕ: − 5 < 𝑥 < 5}
7) List all values of the following set {𝑥|𝑥 ∈ ℤ: − 4 < 𝑥 ≤ 2
8) List the elements in the set {x x is a natural number between 5 and 9}.
9) List the elements in the set
Practice!
10)Set A is the set of all positive integers less than 12.
a) List all of the elements in A.
b) Find n(A).
11)Name the set(s) of numbers to which -14 is an element.
A. Rational
B. Integer
12)Name the set(s) of numbers to which
A. Rational
D. Natural
1
is an element.
2
B. Integer
13)Name the set(s) of numbers to which
A. Rational
C. Irrational
B. Integer
C. Irrational
D. Natural
2 is an element.
C. Irrational
D. Natural
C. { 0, 9,5, 3 }
1
D. { 0, 2,.5, }
3
C. { 0, 2,.5, 3 }
D. { 0, 2,.5,  }
14)Which set of numbers are all integers?
1
A. { 0, 2,.5, }
3
1
B. { 0,  ,.5, }
3
15)Which set of numbers are all rational?
1
A. { 0, 4,.5, }
3
1
B. { 0,  ,.5, }
3
16)Is .23722 an element of the set of real numbers? Explain why or why not.
17)Consider the following sets:
U = {x : x  , 0 ≤ x ≤ 10}, B = {prime numbers ≤10}, C = {x : x 
a. List out all elements of set U, B and C.
b. Explain, in words, what set U represents.
c. What is n(U)?
d. What is n(B)?
e. What is n(C)?
f. Explain, in words, what set C represents.
g. Are all the elements of C also in U?
, 1 ≤ x ≤6}.
18)Consider the following set
:
a. What is n(U)?
b. Is 10 ∈ 𝑈? Explain why or why not!
2
19)Consider the following set: 𝑈 = {−4, − 3 , 1, 𝜋, 13,26.7,68, 1033 }
a. Which elements of this set are irrational?
b. Which elements of this set are integers?
c. Is 13 ∈ ℚ? Explain why or why not!