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1. CHAPTER 1:
SETS AND REAL NUMBERS

2. 1.1 SETS
•Definition:
•A set is a well-defined collection of distinct objects, called elements or
members.
•Notation:
•Sets are usually denoted by capital (uppercase) letters (e.g., A, B, C) and
elements by lowercase letters.
Example:
•A = {a,b,c,d}
•B = {1,2,3,4}
01

3. 02
1.1 SETS
Exercise!
• Write the sets described below using the roster method:
1. The set of vowels in the English alphabet.
2. The set of natural numbers less than 6.
3. The set of all odd numbers between 1 and 10.

4. 03
1.2 DESCRIBING SETS
Two common ways to describe a set:
• Roster Method: List all elements.
• Set-builder Method: Use a rule or property.
Example:
1. Describe the following sets using set-builder notation: B ={2,4,6,8,10}
•Answer: B = {x∣x = 2n, n∈N , 1 ≤ n ≤ 5}
2. Write in roster and in set-builder method: Set A is the set of all multiples of 3
less than 15.
Answer: Roster Method: A = {3,6,9,12}
Set-builder Method: A = {x∣x = 3n, n∈N, x < 15}

5. 04
1.2 DESCRIBING SETS
Exercise!
1. Convert the following sets from roster to set-builder form:
a. {1,4,9,16,25}
b. {2,4,8,16}
2. Write the roster of the set described by {x∣x ∈ Z, −3 ≤ x ≤ 3}.

6. 05
1.3 SUBSETS AND EQUAL SETS
• Subset: A ⊆ B means every element in A is in B.
• Proper subset: A ⊂ B means A ⊆ B but A ≠ B.
• Improper subset: Not a proper subset.
• Trivial subset: A set is either the empty set or the set itself.
• Non-trivial subset: A set that contain some, but not all, of the original
set's elements. All other subsets that are neither empty nor the original set.
• Example: Determine the proper, improper, trivial, and non-trivial subsets of A = {2, 3, 4}.
Answer: Proper subset: { }, {2}, {3}, {4}, {2,3}, {2,4}, {3,4}
Improper subset: {2, 3, 4}
Trivial subset: { }, {2, 3, 4},
Non-trivial subset: {2}, {3}, {4}, {2,3}, {2,4}, {3,4}

7. 06
1.3 SUBSETS AND EQUAL SETS
• Equal sets: A = B means both sets have exactly the same elements.
Example:{1,2}={2,1}
Note: equal sets do not depend on order; order does not matter
• Power set: The set of all subsets of a set A, denoted P(A).
• Example: Power set of {1,2}:
P={∅,{1},{2},{1,2}}.

8. 07
1.3 SUBSETS AND EQUAL SETS
Exercises!
Given A={2,3}, B={1,2,3,4,5}, answer:
1. Is A⊆B?
2. List all subsets of A and determine if it is proper, improper, trivial, and/or
non-trivial.
3. Find the power set of A.
4. Are {a,b,c} and {b,a,c} equal? Explain.

9. 08
1. 4 OPERATIONS ON SETS
•Union: A∪B = {x∣x∈A or x∈B}.
•Intersection: A∩B = {x∣x∈A and x∈B}.
•Difference: A−B = {x∣x∈A and x∉B}.
•Complement: If U is the universal set, then A’ = U−A.
Example:
Let A={1,2,3,4}, B={3,4,5,6}, U={1,2,3,4,5,6,7}.
1) A∪B={1,2,3,4,5,6}
2) A∩B={3,4}
3) A−B={1,2}
4) B’={1,2,7}

10. 09
1. 4 OPERATIONS ON SETS
Exercise!
For X={2,4,6,8}, Y={1,2,3,4}, U: ={1,2,3,4,5,6,7, 8, 9, 10} find:
1. X∪Y
2. X∩Y
3. Y−X
4. X−Y
5. X’
6. Y’

11. 10
1. 5 CARTESIAN PRODUCT
Definition: The Cartesian product A×B={(a,b)∣a∈A, b∈B}.
It is the set of ordered pairs.
•Example:
Given: A={1,2}, B={x,y}.
1. A×B = {(1,x),(1,y),(2,x),(2,y)}
2. B×A =

12. 11
1. 5 CARTESIAN PRODUCT
Activity
• Find:
1. {a,b}×{1,2,3}
2. {0,1}×{0,1}
3. Describe in words the Cartesian product {Red,Blue}×{Shirt,Pants}.

13. 12
1. 6 THE SET OF REAL NUMBERS
Real numbers R include:
Natural numbers: N = {1,2,3,… }
Whole numbers: W={0,1,2,3,… }
Integers: Z={…,−2,−1,0,1,2,… }
Rational numbers Q: numbers that can be written as
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&#3627408515;
where a, b ∈ Z and b ≠ 0
fractions and integers
Irrational numbers Q’: numbers that cannot be written as
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&#3627408515;
, e.g., √2,π)

14. 13
Example:
Classify:
1) 7
2) −5
3)
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&#3627409362;
4) &#3627409363;
1. 6 THE SET OF REAL NUMBERS

15. 14
Exercises!
State which category each of these numbers belong to:
1. 0
2. −3
3. 2.5
4. π
5. Plot −2,
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, 0, &#3627409361; on a number line.
1. 6 THE SET OF REAL NUMBERS

16. 15
Intervals: Set of real numbers between two endpoints.
• Closed interval: [a, b] = {x ∣ a ≤ x ≤ b}
• Open interval: (a, b) = {x ∣ a < x < b}
• Half-open intervals: [a, b) or (a, b]
• Infinite intervals: [a,∞),(−∞,b)
• Represented on number lines
Example
•• (2,5] includes 3, 4, 5 but not 2.
•• (−∞,0) includes all negative numbers.
1. 7 INTERVALS OF REAL NUMBERS

17. 16
Exercises!
Express the solutions of inequalities as intervals:
1) x ≥ 4
2) −3 < x < 2
3) x ≤ −1
1. 7 INTERVALS OF REAL NUMBERS

18. 17
• Closure property: Sum/product of real numbers is a real number.
• Commutative property: a + b = b + a, ab=ba
• Associative property: (a + b) + c = a + (b + c), (ab)c = a(bc)
• Distributive property: a(b + c) = ab + ac
• Identity elements: a + 0 = a, a×1=a
• Inverse elements: a + (−a) = 0, a ∙
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= 1 (where a≠0)
1.8 PROPERTIES OF REAL NUMBERS

19. THANK YOU
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