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Questions and Answers
What is the Cartesian product of two sets A and B?
What is the Cartesian product of two sets A and B?
How is A × B read?
How is A × B read?
What is a property of the Cartesian product?
What is a property of the Cartesian product?
What does a binary relation R between sets A and B represent?
What does a binary relation R between sets A and B represent?
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If A and B are finite sets with |A| = m and |B| = n, how many elements does the Cartesian product A × B have?
If A and B are finite sets with |A| = m and |B| = n, how many elements does the Cartesian product A × B have?
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What can be said about the Cartesian product of infinite sets A and B?
What can be said about the Cartesian product of infinite sets A and B?
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What is a difference between A × B and B × A, when A ≠ B?
What is a difference between A × B and B × A, when A ≠ B?
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What is a common application of the Cartesian product in computer graphics?
What is a common application of the Cartesian product in computer graphics?
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What is the relationship between a binary relation R and the Cartesian product A × B?
What is the relationship between a binary relation R and the Cartesian product A × B?
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Which of the following is NOT a property of the Cartesian product?
Which of the following is NOT a property of the Cartesian product?
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What is the result of the Cartesian product A × (B ∪ C)?
What is the result of the Cartesian product A × (B ∪ C)?
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What is the number of elements in the Cartesian product of three sets A, B, and C?
What is the number of elements in the Cartesian product of three sets A, B, and C?
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In database management systems, what operation is performed using the Cartesian product?
In database management systems, what operation is performed using the Cartesian product?
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What is a consequence of the associativity property of the Cartesian product?
What is a consequence of the associativity property of the Cartesian product?
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Study Notes
Definition
The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
Notation
- A × B is read as "A cross B" or "the Cartesian product of A and B".
- The order of the factors matters, i.e., A × B ≠ B × A (unless A = B).
Properties
- The Cartesian product is associative, i.e., (A × B) × C = A × (B × C).
- The Cartesian product is not commutative, i.e., A × B ≠ B × A (unless A = B).
- The Cartesian product is distributive over union, i.e., A × (B ∪ C) = (A × B) ∪ (A × C).
Relationship With Relations
- A binary relation R between sets A and B can be represented as a subset of the Cartesian product A × B.
- The Cartesian product A × B represents the set of all possible pairs of elements from A and B, while a binary relation R represents a specific subset of these pairs.
Number of Elements
- If A and B are finite sets with |A| = m and |B| = n, then the Cartesian product A × B has |A × B| = m × n elements.
- In general, if A and B are infinite sets, the Cartesian product A × B is also infinite.
Definition of Cartesian Product
- The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
Notation and Key Facts
- A × B is read as "A cross B" or "the Cartesian product of A and B".
- The order of the factors matters, i.e., A × B ≠ B × A (unless A = B).
Properties of Cartesian Product
- The Cartesian product is associative, i.e., (A × B) × C = A × (B × C).
- The Cartesian product is not commutative, i.e., A × B ≠ B × A (unless A = B).
- The Cartesian product is distributive over union, i.e., A × (B ∪ C) = (A × B) ∪ (A × C).
Relationship with Binary Relations
- A binary relation R between sets A and B can be represented as a subset of the Cartesian product A × B.
- The Cartesian product A × B represents the set of all possible pairs of elements from A and B, while a binary relation R represents a specific subset of these pairs.
Number of Elements in Cartesian Product
- If A and B are finite sets with |A| = m and |B| = n, then the Cartesian product A × B has |A × B| = m × n elements.
- In general, if A and B are infinite sets, the Cartesian product A × B is also infinite.
Applications of Cartesian Product
- Used in database management systems to perform joins and combine data from multiple tables
- Applied in statistics to analyze and visualize multivariate data
- Utilized in computer graphics to perform transformations and projections
- Employed in cryptography to define cryptographic protocols
Cartesian Product and Relations
- A binary relation R between sets A and B is a subset of the Cartesian product A × B
- The Cartesian product A × B represents all possible pairs of elements from A and B
- Relations are defined as subsets of the Cartesian product, and operations on relations can be expressed using set operations on the Cartesian product
Properties of Cartesian Product
- The Cartesian product is commutative: A × B = B × A
- The Cartesian product is associative: (A × B) × C = A × (B × C)
- The Cartesian product is distributive over union: A × (B ∪ C) = (A × B) ∪ (A × C)
Number of Elements in Cartesian Product
- The number of elements in the Cartesian product A × B is the product of the number of elements in A and B: |A × B| = |A| × |B|
- This result can be generalized to the Cartesian product of n sets: |A₁ × A₂ × … × An| = |A₁| × |A₂| × … × |An|
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Description
Learn about the Cartesian product of two sets, its notation, properties, and characteristics. Understand the importance of order in the Cartesian product and its applications in mathematics.
