Cartesian Products: Set of Ordered Pairs?

[QuantumP7]

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?"

 

[Outlined]

Question for you: A = {1, 2, 3} and B = {cow, sheep}.Write down A x B.

 

[QuantumP7]

{1 cow, 2 cow, 3 cow; 1 sheep 2 sheep 3 sheep}?

So it's all possible ordered pairs?

 

[Outlined]

{1 cow, 2 cow, 3 cow; 1 sheep 2 sheep 3 sheep}?

correct

So it's all possible ordered pairs?

Your question is more a language thing about the word "possible".

 

[QuantumP7]

Yeah, I've gotten hung up on the semantics of all possible vs. all. But I think that I understand what's going on.

But does anyone know what the Venn diagram of a cartesian product set looks like?

 

[Landau]

correct

Not correct; AxB should consists of ordered pairs.
"1 cow" is not an ordered pair, "(1,cow)" is.

all possible vs. all.

What is or could be the difference?

 

[chiro]

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?"

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.

 

[QuantumP7]

Not correct; AxB should consists of ordered pairs.
"1 cow" is not an ordered pair, "(1,cow)" is.

You're right. My fault.

What is or could be the difference?

I get it now. All Cartesian products = all products. I think that I was just over-thinking the whole thing. Thanks so much!

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.

I see now. Thank you so much.