Cartesian Product in Set Theory Explained Clearly

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 x B = {(a, b)|a ∈ A and b ∈ B}

The Cartesian product of a 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 generalised 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 x 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 x B = {(a, b)|a ∈ A and b ∈ 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 x A = A²

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 x R = R².

Cartesian Product Example

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

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.

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).