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:
“The man who is admired for the ingenuity of his larceny is almost always rediscovering some earlier form of fraud. The basic forms are all known, have all been practicised. The manners of capitalism improve. The morals may not.”
—John Kenneth Galbraith (b. 1908)
“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)