Why is the Cartesian Product S×T Empty When T is Empty?

[rmiller70015]

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?

 

[PeroK]

Can you find anything in $S \times T$?

 

[member 587159]

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?

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]

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?

There is no element t for the set T. And the set theoretical product does not make sense. Except for (∅,∅).

 

[PeroK]

There is no element t for the set T. And the set theoretical product does not make sense. Except for (∅,∅).

$\emptyset \notin \emptyset$

 

[member 587159]

There is no element t for the set T. And the set theoretical product does not make sense. Except for (∅,∅).

Let SxT = {(s,t)|s ∈ S and t ∈ T}
Suppose that SxT ≠ ∅...

Try to find a contradiction, then follows that S x T = ∅

 

[rmiller70015]

Let SxT = {(s,t)|s ∈ S and t ∈ T}
Suppose that SxT ≠ ∅...

Try to find a contradiction, then follows that S x T = ∅

Ok thank you, that makes sense.