Let be a set. A property of is a function
An element is said to have or does not have the property depending on whether or . Any property gives rise in a natural way to the set
and the corresponding http://planetmath.org/node/CharacteristicFunctioncharacteristic function . The identification of with enables us to think of a property of as a 1-ary, or a unary relation on . Therefore, one may treat all these notions equivalently.
Usually, a property of can be identified with a so-called propositional function, or a predicate , where is a variable or a tuple of variables whose values range over . The values of a propositional function is a proposition, which can be interpreted as being either “true” or “false”, so that .
Below are a few examples:
Let . Let be the propositional function “ is divisible by ”. If is the property identified with , then .
Again, let . Let “ is divisible by ” and the corresponding property. Then
which is a subset of . So is a property of .
The reflexive property of a binary relation on can be identified with the propositional function ”, and therefore
which is a subset of . Thus, is a property of .
|Date of creation||2013-03-22 14:01:29|
|Last modified on||2013-03-22 14:01:29|
|Last modified by||drini (3)|