Question One - Proof

Prove that  is a proper subset of

 

A proper subset is a subset that is not equal to the original set.

i.e there are fewer elements in the subset than the original set

 

Proof

.

.

 

Firstly, to prove that  (i.e. All elements of S are in T),

Let .

If , then  for some .

 

Rewriting ,

 

As , it can also be expressed as .

Therefore .

Hence .

 

Secondly, to prove that  (i.e.  is a proper subset),

Let .

If , then  for some .

 

When , then .

 

Now, if , then  for some .

Isolating ,

As  is not an integer,  is not an element of .

Hence .

 

Therefore  is a proper subset of .