Cartesian Product and Ordered Pairs in Set Theory

The Cartesian product of two sets A and B, denoted A B, is the set of all ordered pairs (a, b) where an is in A and b is in B in mathematics, specifically set theory. In terms of set-builder notation are as given below,

$ A \times B = {(a,b) | a \in A \; {\text{and}} \; b \in B}$

The Cartesian product of set of rows and a set of columns can be used to make a table. The cells of the table contain ordered pairs of the type if the Cartesian product rows columns are used (row value, column value). The Cartesian product of ‘n’ sets, also called an n-fold Cartesian product, is a similar concept that can be represented by an n-dimensional array with each element being an n-tuple. A 2-tuple or couple is an ordered pair. The Cartesian product of an indexed family of sets can be defined even more generally. The Cartesian product is a development of René Descartes' definition of analytic geometry, which is further generalized in terms of direct product.

For more understanding let’s discuss one cartesian product of sets example,

Let S & R be two sets such that n(S) = 4 and n(R) = 2. If in the Cartesian product we have (a,1), (b,-1), (c,1), (d, -1). Find S and R, where m, n, x, and y are all distinct.

Solution: S = set of first elements = {a, b, c, d} and R = set of second elements = {1, -1}.

Cartesian Product Definition

The Cartesian product A B between two sets A and B is the set of all possible ordered pairs with the first element from A and a second element from B. The cartesian product formula is given below,

$ A \times B = {(a,b) | a \in A \; {\text{and}} \; b \in B}$

The standard Cartesian coordinates of the plane, where A represents the set of points on the x-axis, B represents the set of points on the y-axis, and A × B represents the XY-plane, are an example.

If A = B, we can denote the Cartesian product of A with itself as $ A \times A = A^2$.

For example, since we can represent the x-axis and the y-axis as the set of real numbers (R), then we can write the XY-plane as $ R \times R = R^2$.

What are The Ordered Pairs?

A set of two things with an order associated with them is referred to as an ordered pair. In most cases, ordered pairings are written in parenthesis (as opposed to curly braces, which are used for writing sets). The element ‘p’ is called the first entry or first component of the ordered pair ( p, q ), and the element ‘q’ is called the second entry or second component of the pair.

Two ordered pairs (p, q) and (x, y) are equal if and only if p = x and q = y. In general,

(p, q) ≠ (x, y).

Steps to Find the Cartesian Product

The cartesian product is also called a cross product. Let us consider two non-empty sets say C = {x, y, z} and D = {1, 2, 3}, these two sets can be represented as shown below,

Hence the cross product of C and D can be found by following the steps:

• Now take the first element from set C i.e ‘x’ and the first element from set D i.e ‘1’.

• These two elements are combined to form an ordered pair (x,1).

• Now take first element from C i.e ‘x’ and second element from D i.e ‘2’, hence ordered pair would be (x,2).

• This process is repeated until all the possible ordered pairs are formed.

• The obtained cartesian product would be sequence of all the ordered pairs.

Cartesian product, CD = {(x,1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3),(z, 1), (z, 2), (z, 3)}

Properties of Cartesian Product

While determining the cross product there are some important properties that are to be followed.

Property 1: The result of cartesian product depends on the order of the pairs, i.e they are non-commutative.

Consider the two sets A and B:

A × B ≠ A × B

A × B = A × B, if and only if A = B.

A × B = ∅, if either A = ∅ or B =∅

Property 2: The rearrangement of the ordered pairs can change the result, hence it doesnot obey associative property. Hence cartesian product is non-associative.

For three sets A, B, and C, (A × B) × C ≠ A × (B × C)

Property 3: The cartesian product is aligned to the distributive property of intersection of the given sets.

For three sets A, B, and C, A × (B ∩ C) = (A × B) ∩ (A × C)

Property 4: The cartesian product is aligned to the distributive property of union of the given sets.

For three sets A, B, and C, A × (B ∪ C) = (A × B) ∪ (A × C)

Cardinality of the Cartesian Product

The total number of elements present in a set is called cardinality of a set. For the set A the cardinal number or cardinality is represented by ‘n(A)’.

Where n(A) = Total number of elements

Whereas the cardinal number of a cartesian product of two sets will be the cross product of cardinal numbers of each set. It can be represented as,

n(A × B) = n(B × A) = n(A) × n(B)

Cartesian Product Solved Example

Example 1: Let A = {1, 2} and B = {1, 2, 3, 4, 5, 6}.

Solution: A × B = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)}

B × A = {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2)}

Therefore, in this case, A × B ≠ B × A.

Hence the Cartesian product is not commutative.

Example 2: Given X = {2,3,4,5} and Y = {3,4,5,6}. Find the following sets,

1. X × Y

2. Y × X

3. X2

4. Y2

Solution:

The given sets are X = {2,3} and Y = {3,4,5,6}.

1. X × Y

By definition, the Cartesian product X × Y contains all feasible ordered pairs ( a, b ) such that a A and b B are the same. As a result, we may write

X × Y = {(2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6)}.

2. Y × X

Similarly, by definition, the Cartesian product Y × X contains all feasible ordered pairs (a, b ) such that a A and b B are the same. As a result, we may write

Y × X = {(3,2), (3,3), (4,2), (4,3), (5,2), (5,3), (6,2), (6,3)}.

3. X2

The cartesian square is defined as the X × X, so we can write as,

X × X = {2,3} × {2,3} = {(2,2), (2,3), (3,2), (3,3)}

4. Y2

The cartesian square is defined as the Y × Y, so we can write as,

Y × Y = {3,4,5,6} × {3,4,5,6} = {(3,3), (3,4), (3,5), (3,6), (4,3), (4,4), (4,5), (4,6), (5,3), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6)}.

Hence it’s solved.

Practise Questions

1. If set A = {3, 5}, then the cardinal number of A × A × A is

1. 4

2. 6

3. 1

4. 8

Ans: Option d

2. If set A = {2, 4, 6}, B = {8, 9} and C = {7, 8}, then find the relation A × (B ∩ C).

1. {(2,7), (4,7), (6,7)}

2. {(2,8), (4,8), (6,8)}

3. {(2,9), (4,9), (6,9)}

4. None of the above

Ans: Option b

Facts

• The given two sets of a cartesian product can be represented in the form of a two-dimensional table. Where the entries will be the elements of each set represented horizontally and vertically.

• In Cartesian product, the first element of an ordered pair will always be chosen from the first set of elements.

Conclusion

The Cartesian product is also termed a cross product, and it is usually implemented in the set theory. Hence the final cartesian product will be the set of all the ordered pairs. It is used in daily life to represent the images of a computer, deck of cards, etc.