Prove that tx.x * P(R - Q) - (P(R) - P(Q)) is well-defined

[DarkSun]

I am not really sure what to make of this Discrete Math question -

Prove that the following expression is well-defined: tx.x * P(R-Q)-(P(R)-P(Q))
* = belongs to...

I know what is P(x) ofcourse, but how do I prove that something is well-defined?

 

[stapel]

Prove that the following expression is well-defined: tx.x * P(R-Q)-(P(R)-P(Q))
* = belongs to...

What is the meaning of "tx.x"? If "tx.x" "belongs to" P(R - Q) - (P(R) - P(Q)), then what is "P(R - Q) - (P(R) - P(Q))"?

I know what is P(x) ofcourse...

What is it?

...but how do I prove that something is well-defined?

The method of proof will probably depend upon the definitions of the elements and relationships, along with whatever tools you may have available.

Please reply with clarification. Thank you!

 

[daon]

I know what is P(x) ofcourse, but how do I prove that something is well-defined?

A function is well defined if it IS a function, and second is DEFINED for all values in its domain. (Some may not consider that second part to be included in the definition of "well-defined" as the definition of Domain people usually encounter is the values for which the function is defined). To show a function is well-defined, assume that two elements are equivilant, say x~y, and show f(x)~f(y). The equivilances depend on the spaces and may not be the same. In general, we'd like to be able to talk about f(x) as if it were a single element of the range.

 

[DarkSun]

Sorry, clarifications:

P(x) means the powerset,
About d, it's the greek symbol (ayota) I am actually not sure what is the right English word for it...
but anyway it stands for "exists exactly one x such that..."
tx.x * P(R - Q) - (P(R) - P(Q))
(* = belongs to...)

Also, It wasn't said the expression is a function (infact I am quite sure it isn't because we only got to the subject of functions later).

About the tools I have, the logic and set theory, so it must lay there.

Thanks.

 

[DarkSun]

Obviously it IS true, taking random sets and checking I can see that it is,
But what does it mean 'show that it's well-defined'? should I prove it formally somehow?

 

[daon]

We still do not know what tx.x means...

For "there exists exactly one" I was sure the symbol was: \(\displaystyle \,\, \exists !\), but when the opportunity came I usually used language to specify uniqueness.

 

[DarkSun]

the symbol you wrote is the same as what I meant.