## Properties of Function

Two properties of functions in general are presented with proofs here.
Below f is a function from a set A to a set B, SA, and T ⊆ B.

Property 1:   f( S T ) = f(S)f(T)

Proof of Property 1:
Proof for f( ST )f(S)f(T) :

Let y be an arbitrary element of f( ST ).
Then there is an element x in ST such that y = f(x).
If x is in S, then y is in f(S). Hence y is in f(S) f(T) .

Similarly y is in f(S)f(T) if x is in T.

Hence if yf( ST ), then yf(S)f(T).
QED
Proof for f(S)f(T) f( ST ) :

Let y be an arbitrary element of f(S)f(T) .
Then y is in f(S) or in f(T).
If y is in f(S), then there is an element x in S such that y = f(x).
Since xS implies xST,  f(x)f( ST ).
Hence yf( ST ).

Similarly yf( ST ) if yf(T) .

QED
Hence Property 1 has been proven.

Property 2:   f( ST )f(S)f(T)

Proof of Property 2:
Let y be an arbitrary element of f( ST ).
Then there is an element x in ST such that y = f(x), that is
there is an element x which is in S and in T, and for which y = f(x) holds.
Hence yf(S) and yf(T), that is yf(S)f(T) .   QED
Note here that the converse of Property 2 does not necessarily hold.
For examle let S = {1}, T = {2}, and f(1) = f(2) = {3}.
Then f( ST ) = f(φ) = φ, while f(S)f(T) = {3}.
Hence f(S)f(T) can not be a subset of f( ST ) giving a counterexample to the converse of Property 2.

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