[Solved]: Is there a difference between $\lambda xy.xy$ and $\lambda x.\lambda y.xy$?

Problem Detail: 

I am currently learning the lambda calculus and was wondering about the following two different kinds of writing a lambda term.

1. $\lambda xy.xy$
2. $\lambda x.\lambda y.xy$

Is there any difference in meaning or the way you apply beta reduction, or are those just two ways to express the same thing?

Especially this definition of pair creation made me wonder:

pair = $\lambda xy.\lambda p.pxy$

Asked By : atticae

Answered By : jmad

These are just differences of notations. $λxyz.t$ is short for $λx.λy.λz.t$. No magic here.

Indeed, $\mbox{pair}=λxyp.pxy$ but you tend to emphasize that $\mbox{pair}\,t\,u$ is a function $λp.ptu$ by changing the way you write the definition. But it is really the same.

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Question Source : http://cs.stackexchange.com/questions/653