CONSTRUCTING AND WRITING PROOFS Prove the following proposit

CONSTRUCTING AND WRITING PROOFS

Prove the following proposition: If p, q elementof Q with p

Solution

Here as p and q both belongs to rational number set Q, we let that

x= p+(q-p)/2 =(2p+q-p)/2 = (p+q)/2

that is the average between two rational numbers p and q , so x mid value of p and q.

Now as it is already given that p <q

so on dividing each side by 2, we get

p/2 < q/2

And to get p in left side, we add p/2 each side, so that it changes as :

p/2+p/2 < q/2 +p/2

or p < (p+q)/2=x

Similarly on adding q/2 each side, we get

p/2+ q/2 < q/2 + q/2

or x=(p+q)/2 <q

so by these two results, we have p < x<q

proved.