29 Questions
What does the cardinality of a set measure?
The number of elements in the set
If A = {1,2,3,4,5} and B = {3,4,5,6,7}, what is A ∩ B?
{3,4,5}
If A = {x, y} and B = {x, y, z}, what is A ⊆ B?
A is a proper subset of B
What does it mean if a set has a cardinality of 0?
The set is the empty set
If A = {1,2,3} and B = {3,4,5}, what is A ∪ B?
{1,2,3,4}
Which set contains only positive numbers?
Natural numbers (N)
What is the cardinality of the set {, {a} , {b} }?
3
Which set includes zero along with all natural numbers?
Whole numbers (W)
What is the correct roster form for the set of real numbers?
{47.3, -12, , …}
Which form of representation is used for defining standard sets like whole numbers?
All of the above
What is the set builder form for the natural numbers (N)?
{x :x is a counting number starting from 1}
Which set is denoted by the symbol Z or I and includes negative natural numbers, zero, and positive natural numbers?
Set Z or I
What is the set builder form for the set of even natural numbers?
{x: x is a natural number, which are divisible by 2}
Which set includes natural numbers that are not divisible by 2?
Set O
What is the roster form for the set of integers?
{..., -2, -1, 0, 1, 2, ...}
Which type of number can be expressed as a fraction with a non-zero denominator?
Rational number
What is the statement form for the set of odd natural numbers?
{1, 3, 5, 7, 9, ...}
Which of the following numbers is rational?
0.09009000900009...
What type of number is √2 / √2?
Rational
Is the number -12 rational or irrational?
Rational, because -12/1 can be expressed as a quotient of two integers.
Why is √3 considered irrational?
Because it cannot be expressed as a quotient of integers.
What category does the number √9 / 25 fall into?
Rational
Why is π/π considered rational?
Because both the numerator and denominator are the same value.
Is √2 a rational number?
No
Which of the following best describes a rational number?
A number that can be simplified to the quotient of two integers
What does the vertical bar '|' in set-builder notation represent?
Such that/where
Which symbol denotes 'is an element of' in set theory?
∈
If a set contains all natural numbers greater than 7, how could it be represented in set-builder notation?
{x | x > 7}
What does '∧' represent in set theory?
Logical AND
Study Notes
Set Basics
- A set can be either finite or infinite.
- Example of a finite set: A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
- Example of an infinite set: Z+ = {1, 2, 3, 4….} (set of positive integers)
Set Notation
- ∈ denotes "is an element of"
- ∉ denotes "is not an element of"
- Example: Yellow ∈ C (Yellow is an element of set C)
- Example: Violet ∉ C (Violet is not an element of set C)
Empty Set
- ∅ denotes an empty set (a set with no elements)
- Example: A = ∅ (set A has no elements)
Cardinality
- The cardinality of a set is a measure of the number of elements in the set.
- Example: The cardinality of set A = {1, 2, 3} is 3.
Standard Sets of Numbers
- Z or I: Set of integers (negative, zero, and positive numbers)
- Example: Z = {……, -3, -2, -1, 0, 1, 2, 3, …….}
- E: Set of even natural numbers (numbers divisible by 2)
- Example: E = {2, 4, 6, 8, …….}
- O: Set of odd natural numbers (numbers not divisible by 2)
- Example: O = {1, 3, 5, 7, 9, …….}
Rational and Irrational Numbers
- Rational numbers can be expressed as a quotient of two integers (a fraction) with a non-zero denominator.
- Irrational numbers cannot be expressed as a quotient of two integers (a fraction) with a non-zero denominator.
- Examples:
- -12 is rational (can be written as -12/1)
- √25 is rational (can be simplified to 5/1)
- 0.09009000900009... is irrational (non-terminating and non-repeating)
- √3/4 is irrational (cannot be expressed as a quotient of two integers)
- √9/25 is rational (can be simplified to 3/5)
Set Builder Notation
- A set-builder notation describes or defines the elements of a set instead of listing the elements.
- Example: {x | x is a counting number less than 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Examples for Sets
- A = {x | x ∈ N ∧ x > 7} = {8, 9, 10, …} (set of all natural numbers greater than 7)
- Let C = {yellow, blue, red} (set of colors)
- The cardinality of C is 3.
Standard Sets of Numbers (continued)
- N: Set of natural numbers (1, 2, 3, …)
- Z: Set of integers (…, -2, -1, 0, 1, 2, …)
- Z+: Set of positive integers (1, 2, 3, …)
- R: Set of real numbers (47.3, -12, π, …)
- Q: Set of rational numbers (1.5, 2.6, -3.8, 15, …)
Test your understanding of sets and elements in discrete mathematics. Explore the concepts of finite and infinite sets, positive integers, set notation, and membership symbols.
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