Group theory · Definition

Cartesian product

setsfunctions

Before you start you need to know

The Cartesian product combines two sets into ordered pairs. It is one of the basic operations used to build relations and functions.

Definition

The Cartesian product of A and B is the set of all ordered pairs whose first coordinate is in A and whose second is in B.

\[A \times B = \{(a,b): a\in A,\ b\in B\}\]

If \(A\) and \(B\) are finite, \(|A \times B| = |A|\cdot|B|\). Also \(A \times \emptyset = \emptyset\), because there is no element of \(\emptyset\) to form a pair with. In general \(A \times B \neq B \times A\), except when \(A = B\) or one of them is empty.

Let \(A=\{1,2\}\) and \(B=\{x,y\}\). Then \(A \times B=\{(1,x),(1,y),(2,x),(2,y)\}\), with 4 pairs. By contrast, \(B \times A=\{(x,1),(x,2),(y,1),(y,2)\}\) is a different set, even though it has the same size.

Typical mistake: writing \((b,a)\) as a pair of \(A \times B\) when \(a\in A\) and \(b\in B\). The order of the coordinates matters. Do not add either: the size is \(|A|\cdot|B|\), not \(|A|+|B|\).

Think about it

If A = {1,2,3} and B = {x,y}, how many pairs does A × B have?

6, because 3 · 2 = 6

Exercises 10

  1. What does A × B contain?
  2. If |A| = 2 and |B| = 3, what is |A × B|?
  3. In (a,b) ∈ A × B, where is a?
  4. In general, do (a,b) and (b,a) represent the same pair?
  5. What happens with A × ∅?
  6. If |A| = 3 and |B| = 2, how many pairs does A × B have?
  7. If A × B = ∅, then necessarily...
  8. When is (a,b) = (c,d)?
  9. Select the only pair that belongs to \({1,2} \times \{a,b\}\).
  10. In general, what is the relation between A × B and B × A?

References

  • I. N. Herstein, Topics in Algebra, 2.ª edición, John Wiley & Sons, 1975.