Math Problem Statement
P2: (b) A database is a collection of records that are manipulated by a computer. The relational database model is based on the concept of an n-ary relation. Let A = {a, b}, B = {1, 2} and C = {x,y} be given sets. If R is a subset of A × B × C then it is called R an n-ary relation. An n-ary relation can be represented by a table or a set of ordered n-tuples. Find (i) A × B × C (ii) (A×B) ×C and its cardinality.
Solution
Let's break down the question and solve each part step by step.
Given Sets:
Part (i):
This represents the Cartesian product of the sets , , and . The Cartesian product consists of all possible ordered triples where:
Let's list all combinations:
So, contains 8 elements.
Part (ii):
Now, we need to compute . This involves first computing the Cartesian product , and then taking the product of this result with .
- First, compute :
- Now, compute :
This consists of all possible ordered pairs where:
Let's list all combinations:
So, also contains 8 elements.
Cardinality of :
The cardinality of is 8, as there are 8 elements in the set.
Summary of Results:
- has 8 elements.
- also has 8 elements.
Do you need more details or have any questions on this?
Related Questions:
- What is the difference between a Cartesian product and a relational database model?
- How does the concept of cardinality relate to database tables?
- How would the result change if we added more elements to set ?
- What would happen if we used a different number of sets for the Cartesian product (e.g., )?
- How does the order of the sets in a Cartesian product affect the resulting tuples?
Tip:
Always remember that the cardinality of a Cartesian product is the product of the cardinalities of the individual sets: .
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Math Problem Analysis
Mathematical Concepts
Set Theory
Cartesian Product
Relations
Cardinality
Formulas
A × B × C = {(a, 1, x), (a, 1, y), (a, 2, x), (a, 2, y), (b, 1, x), (b, 1, y), (b, 2, x), (b, 2, y)}
(A × B) × C = {((a, 1), x), ((a, 1), y), ((a, 2), x), ((a, 2), y), ((b, 1), x), ((b, 1), y), ((b, 2), x), ((b, 2), y)}
Theorems
Cardinality of a Cartesian Product: |A × B × C| = |A| × |B| × |C|
Suitable Grade Level
Grades 10-12
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