Meaning of type Set = Int => Boolean in Scala

Question

I do not understand why Set defined in this way produces these results.

My understanding is that Set is just a function which takes an int and return boolean.

Can someone explain me why I get this result using set?

I think this is a short way to express the function in Scala but I am new to this language and I do not understand how it works.

object sets {
  type Set = Int => Boolean

    var a=Set(3)                              //> a  : scala.collection.immutable.Set[Int] = Set(3)
    a(2)                                      //> res0: Boolean = false
    a(3)                                      //> res1: Boolean = true
    a(1)                                      //> res2: Boolean = false
}

Answer 1

I agree what izstas said they are different things. But "type Set = Int => Boolean" has effect. In the following steps in scala console, after "type Set = Int => Boolean" the error disappears.

scala> def func(s:Set) = 1
<console>:10: error: type Set takes type parameters
       def func(s:Set) = 1
                  ^

scala> def func(s:Set[Int]) = 1
func: (s: Set[Int])Int                      <---- specify the type in the set

scala> type Set = Int => Boolean             
defined type alias Set

scala> def func(s:Set) = 1                   <---- no error
func: (s: Set)Int

scala> def getSet(s:Set) = s
getSet: (s: Set)Set

scala> getSet(Set(1,2))
res0: Set = Set(1, 2)

scala> Set(1,2)
res1: scala.collection.immutable.Set[Int] = Set(1, 2)

scala> val t=Set(1,2)
t: scala.collection.immutable.Set[Int] = Set(1, 2)

Answer 2

The type you defined in type Set = Int => Boolean and the object you created in var a=Set(3) are actually not connected to each other. Even this works:

scala> type Set = String
defined type alias Set

scala> val a = Set(3)
a: scala.collection.immutable.Set[Int] = Set(3)

When you call Set(3), you call apply method on the object Set. If you add new keyword, it will take your type alias into account:

scala> val b = new Set()
b: String = ""

a(2), a(3), a(1) work because Set[A], actually, does implement A => Boolean function trait and apply method is equivalent to contains: http://www.scala-lang.org/api/2.10.3/index.html#scala.collection.immutable.Set