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QuantumP7
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Is the cartesian product [tex] (A \times B) [/tex] the set of ALL POSSIBLE ordered pairs [tex] (a, b) [/tex] 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?"
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 [tex] (A \times B) [/tex] the set of ALL POSSIBLE ordered pairs [tex] (a, b) [/tex] 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.
A Cartesian product is a mathematical operation that combines two sets to create a new set of ordered pairs. The resulting set contains all possible combinations of elements from the two original sets.
A Cartesian product is typically represented using the notation A x B, where A and B are the two sets being combined. For example, if A = {1, 2} and B = {a, b}, then A x B = {(1, a), (1, b), (2, a), (2, b)}.
A Cartesian product is a specific type of cross product, which is a more general mathematical operation that combines any number of sets to create a new set. A Cartesian product specifically combines two sets, while a cross product can combine more than two sets.
Cartesian products are useful in mathematics and science because they allow us to visualize and analyze relationships between different sets of data. They are also used in various mathematical operations, such as matrix multiplication and combinatorics.
Yes, a Cartesian product can be empty if one or both of the original sets are empty. For example, if A = {1, 2} and B = {}, then A x B = {} (an empty set). This is because there are no possible combinations of elements from A and B when B is empty.