Basic Properties
Let A, B, C, and D be sets.
The Cartesian product A × B is not commutative,
because the ordered pairs are reversed except if at least one condition is satisfied:
- A is equal to B, or
- A or B is an empty set.
For example:
- A = {1,2}; B = {3,4}
- A × B = {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)}
- B × A = {3,4} × {1,2} = {(3,1), (3,2), (4,1), (4,2)}
- A = B = {1,2}
- A × B = B × A = {1,2} × {1,2} = {(1,1), (1,2), (2,1), (2,2)}
- A = {1,2}; B = ∅
- A × B = {1,2} × ∅ = ∅
- B × A = ∅ × {1,2} = ∅
Strictly speaking, the Cartesian product is not associative (unless the above condition occurs).
Read more about this topic: Cartesian Product
Famous quotes containing the words basic and/or properties:
“Scientific reason, with its strict conscience, its lack of prejudice, and its determination to question every result again the moment it might lead to the least intellectual advantage, does in an area of secondary interest what we ought to be doing with the basic questions of life.”
—Robert Musil (1880–1942)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)