The discussion centers on the concept of the Cartesian product of two sets, specifically whether it is defined as the set of all possible ordered pairs (a, b) where a is an element of set A and b is an element of set B, or simply as the set of all ordered pairs. The scope includes theoretical aspects and semantic clarifications related to set theory.
Participants generally agree that the Cartesian product consists of ordered pairs, but there is disagreement regarding the interpretation of "all possible" versus "all" ordered pairs, as well as the commutative property of Cartesian products.
There are unresolved semantic issues regarding the definitions of "all possible" and "all" in relation to ordered pairs, and the discussion reflects varying interpretations of these terms.
QuantumP7 said:{1 cow, 2 cow, 3 cow; 1 sheep 2 sheep 3 sheep}?
So it's all possible ordered pairs?
Not correct; AxB should consists of ordered pairs.Outlined said:correct
What is or could be the difference?QuantumP7 said:all possible vs. all.
QuantumP7 said:Is the cartesian product (A \times B) the set of ALL POSSIBLE ordered pairs (a, b) such that a is an element of A and b is an element of b, or is it simply the set of "all ordered pairs?"
Landau said:Not correct; AxB should consists of ordered pairs.
"1 cow" is not an ordered pair, "(1,cow)" is.
What is or could be the difference?
chiro said:If I understand you about trying to say that for example (1,cow) and (cow,1) are the same, then that is false. Cartesian products are not in general commutative since A x B takes the element of A and then B in the ordered pair. if A and B are the same set then you will have this property ( (a,b) and (b,a) are part of A x B) but generally this is not the case.