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Questions and Answers
What is closure?
What is closure?
The property of an operation and a set that the performance of the operation on members of the set always yields a member of the set.
What are integers?
What are integers?
Numbers {0, +1, -1, +2, -2,...} designated with ℤ.
What are irrational numbers?
What are irrational numbers?
Real numbers which cannot be written as the ratio of two integers; designated with ℚ'.
What are natural numbers?
What are natural numbers?
What are rational numbers?
What are rational numbers?
What are real numbers?
What are real numbers?
What are whole numbers?
What are whole numbers?
_____ are the rational numbers together with the irrational numbers; designated with ℝ.
_____ are the rational numbers together with the irrational numbers; designated with ℝ.
_____ are numbers of the form {a/b ∣ a,b ∈ ℤ, b ≠ 0} and designated with ℚ.
_____ are numbers of the form {a/b ∣ a,b ∈ ℤ, b ≠ 0} and designated with ℚ.
_____ are numbers {0, +1, -1, +2, -2,...} designated with ℤ.
_____ are numbers {0, +1, -1, +2, -2,...} designated with ℤ.
Numbers {0, 1, 2, 3, 4,...} are called __.
Numbers {0, 1, 2, 3, 4,...} are called __.
____ are numbers {1, 2, 3, 4,...} and designated with ℕ.
____ are numbers {1, 2, 3, 4,...} and designated with ℕ.
____ are real numbers which cannot be written as the ratio of two integers; designated with ℚ'.
____ are real numbers which cannot be written as the ratio of two integers; designated with ℚ'.
____ is the property of an operation and a set that the performance of the operation on members of the set always yields a member of the set.
____ is the property of an operation and a set that the performance of the operation on members of the set always yields a member of the set.
Is $ ext{√2}$ rational or irrational?
Is $ ext{√2}$ rational or irrational?
Is $ ext{π}$ rational or irrational?
Is $ ext{π}$ rational or irrational?
Is $5$ rational or irrational?
Is $5$ rational or irrational?
Is $0.333...$ rational or irrational?
Is $0.333...$ rational or irrational?
Identify the set: {-2, -1, 0, 1, 2}.
Identify the set: {-2, -1, 0, 1, 2}.
Identify the set: {0, 5, 10, 15}.
Identify the set: {0, 5, 10, 15}.
Identify the set: {4, 5, 6, 7}.
Identify the set: {4, 5, 6, 7}.
Identify the set: {0, 8, 9, 10}.
Identify the set: {0, 8, 9, 10}.
Identify the set described: It is closed under addition and multiplication but not closed under subtraction or division.
Identify the set described: It is closed under addition and multiplication but not closed under subtraction or division.
Identify the set described: It is closed under addition and subtraction, multiplication, and division, with the exception of division by 0 which is not defined.
Identify the set described: It is closed under addition and subtraction, multiplication, and division, with the exception of division by 0 which is not defined.
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Study Notes
Closure
- Represents the property of an operation and a set where performing the operation on elements always results in an element of the same set.
Integers
- Includes numbers {0, +1, -1, +2, -2,...}, denoted by ℤ.
Irrational Numbers
- Real numbers that cannot be expressed as a ratio of two integers; designated with ℚ'.
Natural Numbers
- Consist of positive integers {1, 2, 3, 4,...}, represented by ℕ.
Rational Numbers
- Defined as numbers in the form {a/b | a, b ∈ ℤ, b ≠ 0}, including zero and denoted by ℚ.
Real Numbers
- A combination of rational and irrational numbers, represented by ℝ.
Whole Numbers
- Include all non-negative integers {0, 1, 2, 3,...}.
Rational Identification
- Examples of rational numbers include integers and terminating or repeating decimals (e.g., 5 and 0.333...).
Irrational Identification
- Examples of irrational numbers include non-repeating decimals that cannot be expressed as fractions (e.g., √2 and π).
Set Identification
- Sets {−2, −1, 0, 1, 2} represent integers.
- Sets {0, 5, 10, 15} include whole numbers and integers.
- Sets {4, 5, 6, 7} contain natural numbers, whole numbers, and integers.
- Sets {0, 8, 9, 10} consist of whole numbers and integers.
Natural Numbers and Operations
- Natural numbers are closed under addition and multiplication, but not under subtraction or division.
Real Numbers and Operations
- Real numbers form a complete set closed under addition, subtraction, multiplication, and division, with division by zero undefined.
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