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:

  • A={a,b}A = \{a, b\}
  • B={1,2}B = \{1, 2\}
  • C={x,y}C = \{x, y\}

Part (i): A×B×CA \times B \times C

This represents the Cartesian product of the sets AA, BB, and CC. The Cartesian product A×B×CA \times B \times C consists of all possible ordered triples (a,b,c)(a, b, c) where:

  • aAa \in A
  • bBb \in B
  • cCc \in C

Let's list all combinations:

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 \times B \times 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)\}

So, A×B×CA \times B \times C contains 8 elements.

Part (ii): (A×B)×C(A \times B) \times C

Now, we need to compute (A×B)×C(A \times B) \times C. This involves first computing the Cartesian product A×BA \times B, and then taking the product of this result with CC.

  1. First, compute A×BA \times B:

A×B={(a,1),(a,2),(b,1),(b,2)}A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2)\}

  1. Now, compute (A×B)×C(A \times B) \times C:

This consists of all possible ordered pairs ((a,b),c)((a, b), c) where:

  • (a,b)A×B(a, b) \in A \times B
  • cCc \in C

Let's list all combinations:

(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 \times B) \times 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)\}

So, (A×B)×C(A \times B) \times C also contains 8 elements.

Cardinality of (A×B)×C(A \times B) \times C:

The cardinality of (A×B)×C(A \times B) \times C is 8, as there are 8 elements in the set.

Summary of Results:

  1. 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 \times B \times 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)\} has 8 elements.
  2. (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 \times B) \times 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)\} also has 8 elements.

Do you need more details or have any questions on this?


Related Questions:

  1. What is the difference between a Cartesian product and a relational database model?
  2. How does the concept of cardinality relate to database tables?
  3. How would the result change if we added more elements to set AA?
  4. What would happen if we used a different number of sets for the Cartesian product (e.g., A×B×C×DA \times B \times C \times D)?
  5. 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 A×B×CA \times B \times C is the product of the cardinalities of the individual sets: A×B×C=A×B×C|A \times B \times C| = |A| \times |B| \times |C|.

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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