The discussion revolves around the Cartesian product of sets, specifically examining the case where one of the sets, T, is empty. Participants are exploring why the product S×T results in an empty set when T is empty.
The discussion is active, with participants raising questions about the nature of the Cartesian product and exploring logical reasoning. Some guidance has been offered regarding the definition of the product, and there is an ongoing examination of potential contradictions that arise from assuming S×T is non-empty.
Participants are working under the assumption that S is any set while T is specifically defined as the empty set. There is a focus on the implications of this setup for the Cartesian product.
rmiller70015 said:Homework Statement
Let S be any set and T = ∅. What can you say about the set S×T?
Homework Equations
The Attempt at a Solution
The solution is that S×T=∅. I'm not quite sure why this is though. Is it because there isn't anything in T to give an ordered pair so S×T is empty?
There is no element t for the set T. And the set theoretical product does not make sense. Except for (∅,∅).Math_QED said:Well, when we write out the definition of AxB, we have:
AxB = {(a,b) | a ∈ A and b ∈ B}
Now, apply this to your set, what do you notice?
rmiller70015 said:There is no element t for the set T. And the set theoretical product does not make sense. Except for (∅,∅).
rmiller70015 said:There is no element t for the set T. And the set theoretical product does not make sense. Except for (∅,∅).
Ok thank you, that makes sense.Math_QED said:Let SxT = {(s,t)|s ∈ S and t ∈ T}
Suppose that SxT ≠ ∅...
Try to find a contradiction, then follows that S x T = ∅